title

Precalculus Course

description

Learn Precalculus in this full college course. These concepts are often used in programming.
This course was created by Dr. Linda Green, a lecturer at the University of North Carolina at Chapel Hill. Check out her YouTube channel: https://www.youtube.com/channel/UCkyLJh6hQS1TlhUZxOMjTFw
⭐️ Lecture Notes ⭐️
🔗 Part 1 - Functions: http://lindagreen.web.unc.edu/files/2020/08/courseNotesM130_2019S_Part1_Functions.pdf
🔗 Part 2 - Trigonometry: http://lindagreen.web.unc.edu/files/2020/08/courseNotesM130_2019S_Part2_Trig.pdf
🔗 Part 3 - Conic Sections: http://lindagreen.web.unc.edu/files/2020/08/courseNotesM130_2019S_Part3_ConicSections.pdf
🔗 Part 4 - Parametric Equations, Polar Coordinates, and the Difference Quotient: http://lindagreen.web.unc.edu/files/2020/08/courseNotesM130_2019S_Part4_ParametricPolarDiffQuot.pdf
⭐️ Course Contents ⭐️
⌨️ (0:00:00) Functions
⌨️ (0:12:06) Increasing and Decreasing Functions
⌨️ (0:17:35) Maximums and minimums on graphs
⌨️ (0:26:38) Even and Odd Functions
⌨️ (0:36:12) Toolkit Functions
⌨️ (0:43:18) Transformations of Functions
⌨️ (0:55:48) Piecewise Functions
⌨️ (1:00:19) Inverse Functions
⌨️ (1:14:34) Angles and Their Measures
⌨️ (1:22:47) Arclength and Areas of Sectors
⌨️ (1:28:39) Linear and Radial Speed
⌨️ (1:33:02) Right Angle Trigonometry
⌨️ (1:40:38) Sine and Cosine of Special Angles
⌨️ (1:48:41) Unit Circle Definition of Sine and Cosine
⌨️ (1:54:11) Properties of Trig Functions
⌨️ (1:04:50) Graphs of Sine and Cosine
⌨️ (2:11:23) Graphs of Sinusoidal Functions
⌨️ (2:21:36) Graphs of Tan, Sec, Cot, Csc
⌨️ (2:30:29) Graphs of Transformations of Tan, Sec, Cot, Csc
⌨️ (2:39:02) Inverse Trig Functions
⌨️ (2:48:49) Solving Basic Trig Equations
⌨️ (2:55:49) Solving Trig Equations that Require a Calculator
⌨️ (3:05:44) Trig Identities
⌨️ (3:13:16) Pythagorean Identities
⌨️ (3:18:37) Angle Sum and Difference Formulas
⌨️ (3:26:33) Proof of the Angle Sum Formulas
⌨️ (3:31:09) Double Angle Formulas
⌨️ (3:38:39) Half Angle Formulas
⌨️ (3:44:50) Solving Right Triangles
⌨️ (3:51:24) Law of Cosines
⌨️ (4:01:24) Law of Cosines - old version
⌨️ (4:09:44) Law of Sines
⌨️ (4:17:34) Parabolas - Vertex, Focus, Directrix
⌨️ (4:29:24) Ellipses
⌨️ (4:40:33) Hyperbolas
⌨️ (4:54:23) Polar Coordinates
⌨️ (5:01:55) Parametric Equations
⌨️ (5:13:22) Difference Quotient
--
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detail

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triangles.', 'chapters': [{'end': 100.452, 'segs': [{'end': 77.644, 'src': 'embed', 'start': 18.083, 'weight': 0, 'content': [{'end': 21.705, 'text': 'This video introduces functions and their domains and ranges.', 'start': 18.083, 'duration': 3.622}, {'end': 30.57, 'text': 'A function is a correspondence between input numbers, usually the x values, and output numbers,', 'start': 21.725, 'duration': 8.845}, {'end': 36.172, 'text': 'usually the y values that sends each input number to exactly one output number.', 'start': 30.57, 'duration': 5.602}, {'end': 48.656, 'text': 'Sometimes a function is thought of as a rule or machine in which you can feed in x values as input and get out y values as output.', 'start': 37.233, 'duration': 11.423}, {'end': 55.621, 'text': 'So non mathematical example of a function might be the biological mother function,', 'start': 49.757, 'duration': 5.864}, {'end': 61.265, 'text': 'which takes as input any person and gives us output their biological mother.', 'start': 55.621, 'duration': 5.644}, {'end': 73.182, 'text': 'This function satisfies the condition that each input number, object, person in this case gets sent to exactly one output person,', 'start': 62.537, 'duration': 10.645}, {'end': 77.644, 'text': 'because if you take any person, they just have one biological mother.', 'start': 73.182, 'duration': 4.462}], 'summary': 'Introduction to functions and their correspondence between input and output numbers.', 'duration': 59.561, 'max_score': 18.083, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/eI4an8aSsgw/pics/eI4an8aSsgw18083.jpg'}], 'start': 18.083, 'title': 'Functions and domains', 'summary': 'Introduces functions as a correspondence between input and output numbers, with real-life examples and the condition that each input must map to exactly one output.', 'chapters': [{'end': 100.452, 'start': 18.083, 'title': 'Intro to functions and domains', 'summary': 'Introduces the concept of functions, highlighting that a function is a correspondence between input and output numbers, and provides real-life examples to illustrate the concept, emphasizing the condition that each input must map to exactly one output.', 'duration': 82.369, 'highlights': ['A function is a correspondence between input numbers, usually the x values, and output numbers, usually the y values, that sends each input number to exactly one output number.', 'Real-life example: biological mother function, which takes any person as input and gives their biological mother as output, satisfying the condition that each input number gets sent to exactly one output person.', 'Illustration of non-function: using the mother function, which sends each person to their mother, is no longer a function due to situations where some people have more than one mother.']}], 'duration': 82.369, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/eI4an8aSsgw/pics/eI4an8aSsgw18083.jpg', 'highlights': ['A function is a correspondence between input and output numbers, with each input mapping to exactly one output.', 'Real-life example: biological mother function, satisfying the condition of each input mapping to exactly one output.', 'Illustration of non-function: using the mother function, which is no longer a function in certain situations.']}, {'end': 1422.8, 'segs': [{'end': 177.486, 'src': 'embed', 'start': 149.096, 'weight': 0, 'content': [{'end': 157.841, 'text': "For example, if we want to evaluate f of two, we're plugging in two as input for x, either in this equation, or in that equation.", 'start': 149.096, 'duration': 8.745}, {'end': 165.875, 'text': 'Since f of two means two squared plus one f of two is going to equal five.', 'start': 158.562, 'duration': 7.313}, {'end': 171.699, 'text': 'Similarly, f of five means I plug in five for x.', 'start': 166.936, 'duration': 4.763}, {'end': 177.486, 'text': "So that's going to be 5 squared plus 1, or 26.", 'start': 172.523, 'duration': 4.963}], 'summary': 'When evaluating f(2), f(2) = 5. when evaluating f(5), f(5) = 26.', 'duration': 28.39, 'max_score': 149.096, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/eI4an8aSsgw/pics/eI4an8aSsgw149096.jpg'}, {'end': 294.574, 'src': 'embed', 'start': 262.811, 'weight': 1, 'content': [{'end': 266.392, 'text': "That's because the graph of a circle violates the vertical line test.", 'start': 262.811, 'duration': 3.581}, {'end': 270.754, 'text': "You can draw a vertical line and it'll intersect the graph in more than one point.", 'start': 266.613, 'duration': 4.141}, {'end': 274.756, 'text': 'But our graph at left satisfies the vertical line test.', 'start': 272.195, 'duration': 2.561}, {'end': 277.897, 'text': 'Any vertical line intersects the graph in at most one point.', 'start': 274.976, 'duration': 2.921}, {'end': 284.32, 'text': "That means it's a function because every x value will have at most one y value that corresponds to it.", 'start': 278.457, 'duration': 5.863}, {'end': 288.549, 'text': "Let's evaluate g of 2.", 'start': 285.506, 'duration': 3.043}, {'end': 294.574, 'text': "Note that 2 is an x value, and we'll use the graph to find the corresponding y value.", 'start': 288.549, 'duration': 6.025}], 'summary': 'A graph satisfies the vertical line test, making it a function with at most one y value for each x value.', 'duration': 31.763, 'max_score': 262.811, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/eI4an8aSsgw/pics/eI4an8aSsgw262811.jpg'}, {'end': 482.151, 'src': 'embed', 'start': 450.024, 'weight': 2, 'content': [{'end': 459.15, 'text': 'We think about what x values it makes sense to plug into this expression and what x values need to be excluded because they make the algebraic expression impossible to evaluate.', 'start': 450.024, 'duration': 9.126}, {'end': 470.857, 'text': "Specifically, to find the domain of a function, we need to exclude x values that make the denominator zero, since we can't divide by zero.", 'start': 459.79, 'duration': 11.067}, {'end': 482.151, 'text': "We also need to exclude x values that make an expression inside a square root sign negative, since we can't take the square root of a negative number.", 'start': 472.281, 'duration': 9.87}], 'summary': 'Finding the domain involves excluding x values for division by zero and square root of negative numbers.', 'duration': 32.127, 'max_score': 450.024, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/eI4an8aSsgw/pics/eI4an8aSsgw450024.jpg'}, {'end': 779.357, 'src': 'heatmap', 'start': 574.945, 'weight': 0.979, 'content': [{'end': 581.807, 'text': 'So we need to exclude any x values that make 3 minus 2x less than 0.', 'start': 574.945, 'duration': 6.862}, {'end': 589.79, 'text': 'In other words, we can include all x values for which 3 minus 2x is greater than or equal to 0.', 'start': 581.807, 'duration': 7.983}, {'end': 593.871, 'text': 'Solving that inequality gives us 3 is greater than or equal to 2x.', 'start': 589.79, 'duration': 4.081}, {'end': 598.873, 'text': 'In other words, x is less than or equal to 3 halves.', 'start': 594.551, 'duration': 4.322}, {'end': 605.16, 'text': 'I can draw this on the number line, or write it in interval notation.', 'start': 600.819, 'duration': 4.341}, {'end': 609.401, 'text': 'Notice that three halves is included.', 'start': 607.241, 'duration': 2.16}, {'end': 616.343, 'text': "And that's because three minus two x is allowed to be zero, I can take the square root of zero, it's just zero.", 'start': 609.781, 'duration': 6.562}, {'end': 617.343, 'text': "And that's not a problem.", 'start': 616.503, 'duration': 0.84}, {'end': 623.745, 'text': "Finally, let's look at a more complicated function that involves both a square root and a denominator.", 'start': 618.583, 'duration': 5.162}, {'end': 626.145, 'text': 'Now there are two things I need to worry about.', 'start': 624.465, 'duration': 1.68}, {'end': 632.069, 'text': 'I need the denominator to not be equal to zero.', 'start': 627.626, 'duration': 4.443}, {'end': 637.991, 'text': 'And I need the stuff inside the square root sign to be greater than or equal to zero.', 'start': 632.089, 'duration': 5.902}, {'end': 646.094, 'text': 'From our earlier work, we know that the first condition means that x is not equal to three, and x is not equal to one.', 'start': 638.772, 'duration': 7.322}, {'end': 651.035, 'text': 'And the second condition means that x is less than or equal to three halves.', 'start': 646.694, 'duration': 4.341}, {'end': 654.756, 'text': "Let's draw both of those conditions on the number line.", 'start': 652.156, 'duration': 2.6}, {'end': 659.658, 'text': 'x is not equal to three and x is not equal to one.', 'start': 656.697, 'duration': 2.961}, {'end': 664.87, 'text': "means we've got everything except those two dugout points.", 'start': 660.325, 'duration': 4.545}, {'end': 673.519, 'text': 'And the other condition x is less than or equal to three halves means we can have three halves and everything to the left of it.', 'start': 665.831, 'duration': 7.688}, {'end': 679.345, 'text': 'Now to be in our domain and to be legit for our function, we need both of these conditions be true.', 'start': 674.42, 'duration': 4.925}, {'end': 687.155, 'text': "So I'm going to connect these conditions with an and And that means we're looking for numbers on the number line that are colored both red and blue.", 'start': 679.365, 'duration': 7.79}, {'end': 689.517, 'text': "So I'll draw that above in purple.", 'start': 687.515, 'duration': 2.002}, {'end': 696.221, 'text': "So that's everything from three halves to one, I have to dig out one because one was a problem for the denominator.", 'start': 689.977, 'duration': 6.244}, {'end': 701.645, 'text': 'And then I can continue for all the things that are colored both both colors red and blue.', 'start': 696.861, 'duration': 4.784}, {'end': 714.926, 'text': "So my final domain is going to be let's see negative infinity up to, but not including one together with one, but not including it to three halves.", 'start': 702.125, 'duration': 12.801}, {'end': 718.187, 'text': 'and I include three halves since that was colored both red and blue also.', 'start': 714.926, 'duration': 3.261}, {'end': 724.489, 'text': 'In this video, we talked about functions, how to evaluate them and how to find domains and ranges.', 'start': 719.447, 'duration': 5.042}, {'end': 731.031, 'text': 'This video is a brief introduction to increasing and decreasing functions based on their graphs.', 'start': 726.389, 'duration': 4.642}, {'end': 735.169, 'text': "In this first example, I've graphed two lines.", 'start': 732.728, 'duration': 2.441}, {'end': 738.63, 'text': 'The first line is an increasing function.', 'start': 736.329, 'duration': 2.301}, {'end': 746.252, 'text': 'Because as the x values go from left to right, the y values are going up.', 'start': 740.01, 'duration': 6.242}, {'end': 755.555, 'text': 'This can be written more formally by saying if x two is bigger than x one, these are supposed to be x values.', 'start': 748.373, 'duration': 7.182}, {'end': 761.677, 'text': "So that means that x two is some x value that's to the right, so that is bigger than x one.", 'start': 755.595, 'duration': 6.082}, {'end': 767.389, 'text': "So whenever we have an x value, x two that's bigger than x one.", 'start': 763.587, 'duration': 3.802}, {'end': 779.357, 'text': 'increasing means that f of x two has to be bigger than f of x one, so that the y values are getting bigger when the x values are bigger.', 'start': 767.389, 'duration': 11.968}], 'summary': 'Identified domain for 3-2x and a complex function, explained increasing functions based on graph.', 'duration': 204.412, 'max_score': 574.945, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/eI4an8aSsgw/pics/eI4an8aSsgw574945.jpg'}, {'end': 755.555, 'src': 'embed', 'start': 726.389, 'weight': 3, 'content': [{'end': 731.031, 'text': 'This video is a brief introduction to increasing and decreasing functions based on their graphs.', 'start': 726.389, 'duration': 4.642}, {'end': 735.169, 'text': "In this first example, I've graphed two lines.", 'start': 732.728, 'duration': 2.441}, {'end': 738.63, 'text': 'The first line is an increasing function.', 'start': 736.329, 'duration': 2.301}, {'end': 746.252, 'text': 'Because as the x values go from left to right, the y values are going up.', 'start': 740.01, 'duration': 6.242}, {'end': 755.555, 'text': 'This can be written more formally by saying if x two is bigger than x one, these are supposed to be x values.', 'start': 748.373, 'duration': 7.182}], 'summary': 'Introduction to increasing and decreasing functions based on graph analysis.', 'duration': 29.166, 'max_score': 726.389, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/eI4an8aSsgw/pics/eI4an8aSsgw726389.jpg'}, {'end': 909.6, 'src': 'embed', 'start': 883.206, 'weight': 4, 'content': [{'end': 895.593, 'text': "To describe where this function is decreasing, that's for x values between negative 4 and negative 2 and between 4 and 7.", 'start': 883.206, 'duration': 12.387}, {'end': 906.058, 'text': 'I can write this using inequalities as negative 4 less than x less than negative 2, 4 less than x less than 7.', 'start': 895.593, 'duration': 10.465}, {'end': 909.6, 'text': "It's not important whether I use less than or less than or equal to signs here.", 'start': 906.058, 'duration': 3.542}], 'summary': 'Function decreases for x values between -4 to -2 and 4 to 7.', 'duration': 26.394, 'max_score': 883.206, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/eI4an8aSsgw/pics/eI4an8aSsgw883206.jpg'}, {'end': 1176.317, 'src': 'heatmap', 'start': 954.965, 'weight': 0.709, 'content': [{'end': 967.275, 'text': 'open bracket, a cup sign for union and then 4, comma 7 for the decreasing part and the increasing part is negative 2 comma 1..', 'start': 954.965, 'duration': 12.31}, {'end': 971.728, 'text': "I'm going to modify my graph just a little bit now by putting arrows on the end.", 'start': 967.275, 'duration': 4.453}, {'end': 980.212, 'text': 'When I have arrows instead of dots or just hard stops, that signifies that the function continues in the same direction forever.', 'start': 972.308, 'duration': 7.904}, {'end': 986.496, 'text': 'So if I write it like that, then the place where the function is increasing is still the same.', 'start': 980.232, 'duration': 6.264}, {'end': 993.219, 'text': "It's still going to be from x values going from negative 2 to 1.", 'start': 986.956, 'duration': 6.263}, {'end': 996.681, 'text': 'But now the decreasing part of the function extends further.', 'start': 993.219, 'duration': 3.462}, {'end': 998.842, 'text': 'So this section.', 'start': 997.521, 'duration': 1.321}, {'end': 1004.375, 'text': 'These x values extend all the way out to infinity, and we assume that it keeps going, it keeps decreasing here.', 'start': 999.451, 'duration': 4.924}, {'end': 1010.28, 'text': 'So I would write that as 4 infinity.', 'start': 1005.116, 'duration': 5.164}, {'end': 1020.708, 'text': 'And similarly, the left part I would write as negative infinity, negative 2, because the function is going on forever in this direction, too.', 'start': 1010.8, 'duration': 9.908}, {'end': 1023.39, 'text': 'So the x values go all the way to negative infinity on that chunk.', 'start': 1020.748, 'duration': 2.642}, {'end': 1029.867, 'text': 'So, when talking about increasing and decreasing parts of functions based on graphs,', 'start': 1024.847, 'duration': 5.02}, {'end': 1037.29, 'text': 'the thing to remember is that x is going to be heading from left to right.', 'start': 1029.867, 'duration': 7.423}, {'end': 1041.031, 'text': 'As long as you do that increasing just means that y goes up.', 'start': 1037.31, 'duration': 3.721}, {'end': 1044.291, 'text': 'Decreasing means that y goes down.', 'start': 1042.31, 'duration': 1.981}, {'end': 1052.494, 'text': "And if you're describing the intervals, you have to do that in terms of x values.", 'start': 1045.712, 'duration': 6.782}, {'end': 1061.228, 'text': 'This video is about identifying maximums and minimums for functions from their graphs.', 'start': 1054.861, 'duration': 6.367}, {'end': 1064.071, 'text': "We'll start with some definitions.", 'start': 1062.669, 'duration': 1.402}, {'end': 1076.805, 'text': 'A function f of x has an absolute maximum at the x value of x equals c, if the y value at x equals c is as big as it ever gets.', 'start': 1066.153, 'duration': 10.652}, {'end': 1089.094, 'text': "we can write this more precisely by saying that f of c that's the y value at x equals c is bigger than or equal to f of x.", 'start': 1078.427, 'duration': 10.667}, {'end': 1091.055, 'text': "that's the y value at some other x value.", 'start': 1089.094, 'duration': 1.961}, {'end': 1109.352, 'text': 'For all x values in the domain, of f, that biggest y value f of c is called the absolute maximum value for f.', 'start': 1092.376, 'duration': 16.976}, {'end': 1115.078, 'text': 'And the point with an x value and a y value given is called an absolute maximum point.', 'start': 1109.352, 'duration': 5.726}, {'end': 1127.94, 'text': 'If I have a graph of a function, then the absolute maximum value f of c is the highest value that function ever achieves.', 'start': 1116.319, 'duration': 11.621}, {'end': 1133.042, 'text': 'And the absolute maximum point is the point where it achieves that value.', 'start': 1128.38, 'duration': 4.662}, {'end': 1143.087, 'text': "Now it's possible for a function to have more than one absolute maximum point, if there happens to be a tie, where that highest value is achieved.", 'start': 1134.803, 'duration': 8.284}, {'end': 1148.569, 'text': 'But a function can only have most one absolute maximum value.', 'start': 1144.027, 'duration': 4.542}, {'end': 1164.07, 'text': 'a function f of x has an absolute minimum at x equals c, if the y value at x equals c is as small as it ever gets.', 'start': 1150.362, 'duration': 13.708}, {'end': 1176.317, 'text': 'We can write this more precisely by saying that f of c is less than or equal to f of x for all x values in the domain of f.', 'start': 1165.491, 'duration': 10.826}], 'summary': 'The video explains increasing, decreasing sections of a function and identifying maximums and minimums from their graphs.', 'duration': 221.352, 'max_score': 954.965, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/eI4an8aSsgw/pics/eI4an8aSsgw954965.jpg'}], 'start': 100.852, 'title': 'Understanding and analyzing functions', 'summary': 'Covers understanding functions, function notation, domains, ranges, and graph analysis using examples, graphs, and algebraic considerations.', 'chapters': [{'end': 235.414, 'start': 100.852, 'title': 'Understanding functions and function notation', 'summary': 'Explains the concept of functions and function notation, using examples to illustrate how to evaluate functions and the importance of maintaining parentheses when plugging in complex expressions.', 'duration': 134.562, 'highlights': ['The function notation f(x) is used to represent the output value of y, not multiplication, with an example of f(2) resulting in 5 and f(5) resulting in 26.', 'It is important to maintain parentheses when evaluating a function on a complex expression, as demonstrated when evaluating f(a+3) resulting in a^2 + 6a + 10.', 'Functions are described with equations, and the violation of the rule of functions results in a non-function, with the example of input producing more than one possible output.', 'The concept of evaluating a function on a more complicated expression involving other variables is explained, with the example of evaluating f(a+3) resulting in a^2 + 6a + 10.']}, {'end': 434, 'start': 235.414, 'title': 'Understanding functions and their domains', 'summary': 'Explains the concept of functions using graphs, introduces the vertical line test to determine if a graph represents a function, evaluates specific values of a function using a graph, discusses the domain and range of a function, and provides examples of domain and range notation.', 'duration': 198.586, 'highlights': ['The vertical line test is used to determine if a graph represents a function.', 'Evaluating g of 2 results in g of 2 = 3, while g of 5 is undefined.', 'The domain of the function g is between -8 and 4, while the range is between -5 and 3.']}, {'end': 724.489, 'start': 435.276, 'title': 'Finding domain and range of functions', 'summary': 'Discusses finding the domain of functions using algebraic considerations, excluding x values that make the denominator zero or the expression inside a square root sign negative, illustrated through examples.', 'duration': 289.213, 'highlights': ['To find the domain of a function, exclude x values that make the denominator zero and those that make an expression inside a square root sign negative.', 'Excluding x values that make the denominator zero and solving for the inequality 3 - 2x >= 0 allows us to determine the domain for specific functions.', 'Illustrated examples demonstrate how to find the domain for functions with denominators and square root signs, using algebraic considerations to exclude specific x values.']}, {'end': 881.302, 'start': 726.389, 'title': 'Functions and their graphs', 'summary': 'Introduces increasing and decreasing functions based on their graphs, demonstrating how to formally define increasing and decreasing functions, as well as identifying intervals of increasing, decreasing, and constant behavior on a graph.', 'duration': 154.913, 'highlights': ['The chapter introduces increasing and decreasing functions based on their graphs.', 'Demonstrating how to formally define increasing and decreasing functions.', 'Identifying intervals of increasing, decreasing, and constant behavior on a graph.']}, {'end': 1422.8, 'start': 883.206, 'title': 'Identifying function behavior', 'summary': 'Discusses the behavior of a function, including its increasing and decreasing intervals, absolute maximum and minimum values, and local maximum and minimum points, using inequalities, interval notation, and graph analysis.', 'duration': 539.594, 'highlights': ['The function is decreasing for x values between negative 4 and negative 2 and between 4 and 7, indicated by strict less than signs at the endpoints and expressed in inequalities as -4 < x < -2, 4 < x < 7.', 'The function is increasing for x values between -2 and 1, described by the inequality -2 < x < 1 and represented in interval notation as [-2, 1].', "The concept of absolute maximum and minimum values for a function is introduced, defining the conditions and characteristics of these points in the context of a graph and their relation to the function's behavior."]}], 'duration': 1321.948, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/eI4an8aSsgw/pics/eI4an8aSsgw100852.jpg', 'highlights': ['The function notation f(x) represents the output value of y, with f(2) resulting in 5 and f(5) resulting in 26.', 'The vertical line test is used to determine if a graph represents a function.', 'To find the domain of a function, exclude x values that make the denominator zero and those that make an expression inside a square root sign negative.', 'The chapter introduces increasing and decreasing functions based on their graphs.', 'The function is decreasing for x values between negative 4 and negative 2 and between 4 and 7, indicated by strict less than signs at the endpoints and expressed in inequalities as -4 < x < -2, 4 < x < 7.']}, {'end': 2618.722, 'segs': [{'end': 1500.972, 'src': 'embed', 'start': 1459.967, 'weight': 0, 'content': [{'end': 1466.513, 'text': 'in an open interval around x equals four.', 'start': 1459.967, 'duration': 6.546}, {'end': 1475.365, 'text': 'local maximum and minimum values can also be called relative maximum and minimum values.', 'start': 1468.641, 'duration': 6.724}, {'end': 1485.492, 'text': 'Please pause the video for a moment to mark all local maximum minimum points on the graph of this function,', 'start': 1478.927, 'duration': 6.565}, {'end': 1487.613, 'text': 'as well as all absolute max and min points.', 'start': 1485.492, 'duration': 2.121}, {'end': 1500.972, 'text': 'the function has a local max point here at the point with coordinates approximately say 2.7 3.3.', 'start': 1490.353, 'duration': 10.619}], 'summary': 'Identified local max point at (2.7, 3.3) around x=4.', 'duration': 41.005, 'max_score': 1459.967, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/eI4an8aSsgw/pics/eI4an8aSsgw1459967.jpg'}, {'end': 1573.081, 'src': 'embed', 'start': 1526.601, 'weight': 1, 'content': [{'end': 1530.602, 'text': "So there's no local min point here, and no absolute min point.", 'start': 1526.601, 'duration': 4.001}, {'end': 1538.088, 'text': 'The point here with coordinates 04 is considered by some sources to be a local max point.', 'start': 1531.462, 'duration': 6.626}, {'end': 1544.574, 'text': "But by other sources, it's not considered a local max point, since the function is not defined in an open interval around zero.", 'start': 1538.308, 'duration': 6.266}, {'end': 1548.017, 'text': 'Turning our attention to absolute maximum points.', 'start': 1545.635, 'duration': 2.382}, {'end': 1553.762, 'text': "Well, there's no absolute max point since the function keeps going up and up and up forever.", 'start': 1548.077, 'duration': 5.685}, {'end': 1565.575, 'text': "there's also no absolute min point, because the function just keeps going down and down a little by little and never actually achieves a lowest value.", 'start': 1555.267, 'duration': 10.308}, {'end': 1573.081, 'text': "Now if we're talking about local max and min values, that's just the y values of the local max and min points.", 'start': 1566.836, 'duration': 6.245}], 'summary': 'No absolute max/min points; 04 may be a local max point but not universally agreed upon; function keeps going up and down indefinitely.', 'duration': 46.48, 'max_score': 1526.601, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/eI4an8aSsgw/pics/eI4an8aSsgw1526601.jpg'}, {'end': 1859.384, 'src': 'embed', 'start': 1833.513, 'weight': 4, 'content': [{'end': 1840.536, 'text': 'Please pause the video for a moment and decide which of these graphs are symmetric with respect to the x-axis, the y-axis, and the origin.', 'start': 1833.513, 'duration': 7.023}, {'end': 1843.277, 'text': 'Some graphs may have more than one type of symmetry.', 'start': 1841.236, 'duration': 2.041}, {'end': 1854.081, 'text': 'Graph A is symmetric with respect to the origin, because if you rotate it by 180 degrees, that is, you turn it upside down, it looks exactly the same.', 'start': 1844.518, 'duration': 9.563}, {'end': 1859.384, 'text': "It does not have any mirror symmetry, so it's not symmetric with respect to the x-axis or the y-axis.", 'start': 1854.822, 'duration': 4.562}], 'summary': 'Identify symmetry of graphs a with respect to x-axis, y-axis, and origin.', 'duration': 25.871, 'max_score': 1833.513, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/eI4an8aSsgw/pics/eI4an8aSsgw1833513.jpg'}, {'end': 1907.947, 'src': 'embed', 'start': 1879.931, 'weight': 5, 'content': [{'end': 1884.732, 'text': 'Graph C is symmetric with respect to the y axis, but has no other symmetry.', 'start': 1879.931, 'duration': 4.801}, {'end': 1890.414, 'text': 'And graph D is symmetric with respect to the x axis with no other symmetry.', 'start': 1885.453, 'duration': 4.961}, {'end': 1900.284, 'text': 'We use the words even and odd to describe functions whose graphs have certain kinds of symmetry.', 'start': 1892.742, 'duration': 7.542}, {'end': 1907.947, 'text': 'A function is even if its graph is symmetric with respect to the y-axis.', 'start': 1901.165, 'duration': 6.782}], 'summary': 'Graph c is symmetric to y-axis, while graph d is symmetric to x-axis.', 'duration': 28.016, 'max_score': 1879.931, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/eI4an8aSsgw/pics/eI4an8aSsgw1879931.jpg'}, {'end': 2183.663, 'src': 'embed', 'start': 2161.112, 'weight': 3, 'content': [{'end': 2169.114, 'text': "There's no word for functions that are symmetric with respect to the x-axis, and that's because if your graph is symmetric with respect to the x-axis,", 'start': 2161.112, 'duration': 8.002}, {'end': 2170.174, 'text': "it's not going to be a function.", 'start': 2169.114, 'duration': 1.06}, {'end': 2176.696, 'text': 'This video gives the graphs of some commonly used functions that I call the toolkit functions.', 'start': 2171.855, 'duration': 4.841}, {'end': 2181.362, 'text': 'The first function is the function y equals x.', 'start': 2177.779, 'duration': 3.583}, {'end': 2183.663, 'text': "Let's plot a few points on the graph of this function.", 'start': 2181.362, 'duration': 2.301}], 'summary': 'Toolkit functions explained, including y=x graph.', 'duration': 22.551, 'max_score': 2161.112, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/eI4an8aSsgw/pics/eI4an8aSsgw2161112.jpg'}], 'start': 1422.8, 'title': 'Functions and symmetry', 'summary': 'Covers local maxima and minima, symmetry concepts with x-axis, y-axis, and origin, toolkit functions, and their symmetries. it provides examples and verifications, emphasizing local extrema points for a given function.', 'chapters': [{'end': 1601.7, 'start': 1422.8, 'title': 'Classifying local maxima and minima', 'summary': 'Explains the concept of local maxima and minima, discussing the criteria for identifying and distinguishing them, emphasizing that for the given function, it has a local maximum at (2.7, 3.3) and a local minimum at (2,2). it also clarifies that no absolute maximum or minimum exists for the function.', 'duration': 178.9, 'highlights': ['The function has a local maximum at the point with coordinates approximately (2.7, 3.3), and a local minimum at the point with coordinates (2, 2).', 'The function has no absolute maximum or minimum points as it continuously goes up and down without reaching a highest or lowest value.', 'Some sources consider the point at (0, 4) to be a local maximum, while others do not, due to the function not being defined in an open interval around zero.']}, {'end': 2138.335, 'start': 1603.462, 'title': 'Symmetry of graphs', 'summary': 'Explains the concepts of symmetry with respect to the x-axis, y-axis, and the origin in graphs, and provides examples and algebraic verifications of even and odd functions. it also illustrates the determination of graph symmetries and provides examples of graphs with different types of symmetry.', 'duration': 534.873, 'highlights': ['Graph B exhibits symmetry with respect to the x-axis, y-axis, and the origin, while Graph A and C exhibit symmetry with respect to the origin and y-axis, and the x-axis respectively.', 'A function is even if its graph is symmetric with respect to the y-axis, and a function is odd if its graph is symmetric with respect to the origin.', 'The function f of x equals x squared plus 3 is even, verified by checking that f of negative x equals f of x, and the function f of x equals 5x minus 1 over x is odd, verified by checking if f of negative x equals negative of f of x.']}, {'end': 2618.722, 'start': 2139.976, 'title': 'Toolkit functions and symmetry', 'summary': 'Discusses the graphs and symmetries of toolkit functions, including the even and odd symmetries of functions, and provides examples of commonly used functions such as linear, quadratic, cubic, square root, absolute value, exponential, hyperbola, and their transformations.', 'duration': 478.746, 'highlights': ['The toolkit functions include linear, quadratic, cubic, square root, absolute value, exponential, and hyperbola functions, each with specific symmetries and shapes.', 'Functions symmetric with respect to the y-axis are even functions, while those symmetric with respect to the origin are odd functions.', 'Examples of functions exhibiting even and odd symmetries are provided, including the cubic and square root functions, demonstrating their mirror and rotational symmetries.']}], 'duration': 1195.922, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/eI4an8aSsgw/pics/eI4an8aSsgw1422800.jpg', 'highlights': ['The function has a local maximum at the point with coordinates approximately (2.7, 3.3), and a local minimum at the point with coordinates (2, 2).', 'The function has no absolute maximum or minimum points as it continuously goes up and down without reaching a highest or lowest value.', 'Some sources consider the point at (0, 4) to be a local maximum, while others do not, due to the function not being defined in an open interval around zero.', 'The toolkit functions include linear, quadratic, cubic, square root, absolute value, exponential, and hyperbola functions, each with specific symmetries and shapes.', 'Graph B exhibits symmetry with respect to the x-axis, y-axis, and the origin, while Graph A and C exhibit symmetry with respect to the origin and y-axis, and the x-axis respectively.', 'A function is even if its graph is symmetric with respect to the y-axis, and a function is odd if its graph is symmetric with respect to the origin.']}, {'end': 3574.035, 'segs': [{'end': 3482.459, 'src': 'heatmap', 'start': 3019.25, 'weight': 0, 'content': [{'end': 3027.337, 'text': 'y equals the square root of x minus 2, those numbers affect the y values and result in vertical motions like we saw.', 'start': 3019.25, 'duration': 8.087}, {'end': 3030.38, 'text': 'These motions are in the direction you expect.', 'start': 3028.338, 'duration': 2.042}, {'end': 3034.083, 'text': 'So subtracting 2 moves us down by 2.', 'start': 3030.74, 'duration': 3.343}, {'end': 3038.365, 'text': 'If we were adding 2 instead, that would move us up by 2.', 'start': 3034.083, 'duration': 4.282}, {'end': 3040.526, 'text': 'Numbers on the inside of the function.', 'start': 3038.365, 'duration': 2.161}, {'end': 3041.667, 'text': "that's like our example.", 'start': 3040.526, 'duration': 1.141}, {'end': 3049.23, 'text': 'y equals the square root of quantity x minus 2, those affect the x values and result in a horizontal motion.', 'start': 3041.667, 'duration': 7.563}, {'end': 3052.532, 'text': 'These motions go in the opposite direction from what you expect.', 'start': 3049.891, 'duration': 2.641}, {'end': 3057.474, 'text': 'Remember, the minus 2 on the inside actually shifted our graph to the right.', 'start': 3053.212, 'duration': 4.262}, {'end': 3062.457, 'text': 'If it had a plus 2 on the inside, that would actually shift our graph to the left.', 'start': 3058.115, 'duration': 4.342}, {'end': 3066.526, 'text': 'adding results in a shift.', 'start': 3064.364, 'duration': 2.162}, {'end': 3073.972, 'text': 'those are called translations, but multiplying like something like y equals three times the square root of x.', 'start': 3066.526, 'duration': 7.446}, {'end': 3076.774, 'text': 'that would result in a stretch or shrink.', 'start': 3073.972, 'duration': 2.802}, {'end': 3085.921, 'text': 'In other words, if I start with the square root of x and then when a graph y equals three times the square root of x,', 'start': 3078.195, 'duration': 7.726}, {'end': 3091.526, 'text': 'that stretches my graph vertically by a factor of three like this', 'start': 3085.921, 'duration': 5.605}, {'end': 3100.697, 'text': 'If I want to graph y equals 1 third times the square root of x, that shrinks my graph vertically by a factor of 1 third.', 'start': 3093.194, 'duration': 7.503}, {'end': 3104.898, 'text': 'Finally, a negative sign results in reflection.', 'start': 3102.077, 'duration': 2.821}, {'end': 3114.622, 'text': 'For example, if I start with, a graph of y equals the square root of x and then when a graph y equals the square root of negative x,', 'start': 3105.679, 'duration': 8.943}, {'end': 3120.284, 'text': "that's going to do a reflection in the horizontal direction, because the negative is on the inside of the square root sign.", 'start': 3114.622, 'duration': 5.662}, {'end': 3125.98, 'text': 'A reflection in the horizontal direction means a reflection across the y axis.', 'start': 3121.638, 'duration': 4.342}, {'end': 3135.223, 'text': 'If instead I want to graph y equals negative the square root of x, that negative sign on the outside means a vertical reflection,', 'start': 3127.92, 'duration': 7.303}, {'end': 3137.284, 'text': 'a reflection across the x axis.', 'start': 3135.223, 'duration': 2.061}, {'end': 3145.007, 'text': 'Pause the video for a moment and see if you can describe what happens in these four transformations.', 'start': 3140.305, 'duration': 4.702}, {'end': 3150.161, 'text': "In the first example, we're subtracting four on the outside of the function.", 'start': 3146.38, 'duration': 3.781}, {'end': 3153.862, 'text': 'Adding and subtracting means a translation or shift.', 'start': 3151.181, 'duration': 2.681}, {'end': 3159.563, 'text': "And since we're on the outside of the function affects the y values that's moving us vertically.", 'start': 3154.902, 'duration': 4.661}, {'end': 3166.984, 'text': 'So this transformation should take the square root of graph and move it down by four units.', 'start': 3160.483, 'duration': 6.501}, {'end': 3169.325, 'text': 'That would look something like this.', 'start': 3167.865, 'duration': 1.46}, {'end': 3177.176, 'text': "In the next example, we're adding 12 on the inside, that's still a translation.", 'start': 3171.874, 'duration': 5.302}, {'end': 3179.377, 'text': "But now we're moving horizontally.", 'start': 3177.557, 'duration': 1.82}, {'end': 3187.841, 'text': "And so since we go the opposite direction we expect, we are going to go to the left by 12 units, that's going to look something like this.", 'start': 3179.938, 'duration': 7.903}, {'end': 3195.185, 'text': "In the next example, we're multiplying by three and introducing a negative sign, both on the outside of our function.", 'start': 3189.722, 'duration': 5.463}, {'end': 3198.745, 'text': "outside our function means we're affecting the y values.", 'start': 3196.323, 'duration': 2.422}, {'end': 3207.212, 'text': "So and multiplication means we're stretching by a factor of three, the negative sign means we reflect in the vertical direction.", 'start': 3199.245, 'duration': 7.967}, {'end': 3212.997, 'text': "Here's stretching by a factor of three vertically, before I apply the minus sign.", 'start': 3208.753, 'duration': 4.244}, {'end': 3216.8, 'text': 'And now the minus sign reflects in the vertical direction.', 'start': 3213.497, 'duration': 3.303}, {'end': 3227.032, 'text': "Finally, in this last example, we're multiplying by one quarter on the inside of our function, we know that multiplication means stretch or shrink.", 'start': 3218.186, 'duration': 8.846}, {'end': 3233.537, 'text': "And since we're on the inside, it's a horizontal motion, and it does the opposite of what we expect.", 'start': 3228.313, 'duration': 5.224}, {'end': 3242.404, 'text': "So instead of shrinking by a factor of one fourth horizontally, it's actually going to stretch by the reciprocal, a factor of four horizontally.", 'start': 3234.118, 'duration': 8.286}, {'end': 3244.969, 'text': "that'll look something like this.", 'start': 3243.649, 'duration': 1.32}, {'end': 3252.851, 'text': 'Notice that stretching horizontally by a factor of four looks kind of like shrinking vertically by a factor of one half.', 'start': 3246.25, 'duration': 6.601}, {'end': 3256.732, 'text': "And that's actually borne out by the algebra,", 'start': 3254.251, 'duration': 2.481}, {'end': 3263.013, 'text': 'because the square root of one fourth x is the same thing as the square root of one fourth times the square root of x,', 'start': 3256.732, 'duration': 6.281}, {'end': 3266.834, 'text': 'which is the same thing as one half times the square root of x.', 'start': 3263.013, 'duration': 3.821}, {'end': 3277.112, 'text': 'And So now we can see algebraically that a vertical shrink by a factor of one half is the same as a horizontal stretch by a factor of four,', 'start': 3266.834, 'duration': 10.278}, {'end': 3279.213, 'text': 'at least for this function, the square root function.', 'start': 3277.112, 'duration': 2.101}, {'end': 3285.175, 'text': "This video gave some rules for transformations of functions, which I'll repeat below.", 'start': 3280.954, 'duration': 4.221}, {'end': 3291.217, 'text': 'Numbers on the outside correspond to changes in the y values.', 'start': 3286.335, 'duration': 4.882}, {'end': 3294.093, 'text': 'or vertical motions.', 'start': 3292.373, 'duration': 1.72}, {'end': 3301.575, 'text': 'Numbers on the inside of the function affect the x values and result in horizontal motions.', 'start': 3295.494, 'duration': 6.081}, {'end': 3308.357, 'text': 'Adding and subtracting corresponds to translations or shifts.', 'start': 3302.896, 'duration': 5.461}, {'end': 3314.918, 'text': 'Multiplying and dividing by numbers corresponds to stretches and shrinks.', 'start': 3310.177, 'duration': 4.741}, {'end': 3321.1, 'text': 'And putting in a negative sign corresponds to a reflection.', 'start': 3316.459, 'duration': 4.641}, {'end': 3330.32, 'text': 'horizontal reflection, if the negative sign is on the inside, and a vertical reflection, if the negative sign is on the outside.', 'start': 3322.917, 'duration': 7.403}, {'end': 3342.704, 'text': 'Knowing these basic rules about transformations empowers you to be able to sketch graphs of much more complicated functions like y equals three times the square root of x plus two,', 'start': 3331.58, 'duration': 11.124}, {'end': 3346.326, 'text': 'by simply considering the transformations one at a time.', 'start': 3342.704, 'duration': 3.622}, {'end': 3350.147, 'text': 'This video introduces piecewise functions.', 'start': 3347.846, 'duration': 2.301}, {'end': 3360.153, 'text': "a piecewise function is a function who's defined in pieces by two or more different rules that apply for different x values.", 'start': 3352.011, 'duration': 8.142}, {'end': 3370.515, 'text': 'The rule for calculating this function is you calculate negative x squared if x is less than one, but you calculate negative two, x plus three,', 'start': 3361.693, 'duration': 8.822}, {'end': 3372.236, 'text': 'if x is greater than or equal to one.', 'start': 3370.515, 'duration': 1.721}, {'end': 3378.357, 'text': 'So if we want to find f of negative two, well, negative two is less than one.', 'start': 3373.436, 'duration': 4.921}, {'end': 3379.898, 'text': 'So the first rule applies.', 'start': 3378.797, 'duration': 1.101}, {'end': 3387.714, 'text': 'And we compute f of negative two, by plugging in negative two for x in the first rule.', 'start': 3380.898, 'duration': 6.816}, {'end': 3389.056, 'text': "So that's negative four.", 'start': 3387.975, 'duration': 1.081}, {'end': 3396.924, 'text': 'Next, if we want to find f of one, Well, 1 is right on the border in between the two rules.', 'start': 3390.318, 'duration': 6.606}, {'end': 3402.847, 'text': 'but because we have a greater than or equal to here, when x is equal to 1, we apply this rule.', 'start': 3396.924, 'duration': 5.923}, {'end': 3413.213, 'text': "And so we plug 1 into the formula negative 2x plus 3, so that's negative 2 times 1 plus 3, which gives us 1 as our output.", 'start': 3403.487, 'duration': 9.726}, {'end': 3422.898, 'text': 'Finally, if we want to compute f of 3, since 3 is bigger than or equal to 1, the second rule applies, and we plug 3 into that rule.', 'start': 3414.173, 'duration': 8.725}, {'end': 3426.941, 'text': 'that gives us an answer of negative three.', 'start': 3424.4, 'duration': 2.541}, {'end': 3431.303, 'text': 'To graph f, it makes sense to also draw the graph in pieces.', 'start': 3427.901, 'duration': 3.402}, {'end': 3434.584, 'text': "First, I'm going to draw the graph of minus x squared.", 'start': 3431.783, 'duration': 2.801}, {'end': 3439.386, 'text': 'Just x squared would be a parabola opening up.', 'start': 3436.665, 'duration': 2.721}, {'end': 3449.05, 'text': 'So minus x squared is a parabola opening down goes through the points negative one, negative one and one negative one.', 'start': 3439.606, 'duration': 9.444}, {'end': 3451.651, 'text': 'So it looks something like this.', 'start': 3450.41, 'duration': 1.241}, {'end': 3460.02, 'text': "Now I've drawn the whole parabola, but the rule actually only applies when x is less than 1.", 'start': 3454.878, 'duration': 5.142}, {'end': 3469.165, 'text': "So I'll keep the part of the parabola when x is less than 1, and I'll erase the part of the parabola when x is greater than or equal to 1.", 'start': 3460.02, 'duration': 9.145}, {'end': 3475.508, 'text': "I'll leave an open circle here when x equals 1, since that point is not included in this definition either.", 'start': 3469.165, 'duration': 6.343}, {'end': 3482.459, 'text': "Next, I'm going to draw the second piece, the line y equals minus 2x plus 3.", 'start': 3476.556, 'duration': 5.903}], 'summary': 'Understanding graph transformations and piecewise functions for complex function sketching.', 'duration': 72.276, 'max_score': 3019.25, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/eI4an8aSsgw/pics/eI4an8aSsgw3019250.jpg'}, {'end': 3145.007, 'src': 'embed', 'start': 3114.622, 'weight': 3, 'content': [{'end': 3120.284, 'text': "that's going to do a reflection in the horizontal direction, because the negative is on the inside of the square root sign.", 'start': 3114.622, 'duration': 5.662}, {'end': 3125.98, 'text': 'A reflection in the horizontal direction means a reflection across the y axis.', 'start': 3121.638, 'duration': 4.342}, {'end': 3135.223, 'text': 'If instead I want to graph y equals negative the square root of x, that negative sign on the outside means a vertical reflection,', 'start': 3127.92, 'duration': 7.303}, {'end': 3137.284, 'text': 'a reflection across the x axis.', 'start': 3135.223, 'duration': 2.061}, {'end': 3145.007, 'text': 'Pause the video for a moment and see if you can describe what happens in these four transformations.', 'start': 3140.305, 'duration': 4.702}], 'summary': 'Reflections in horizontal and vertical directions across y and x axes described.', 'duration': 30.385, 'max_score': 3114.622, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/eI4an8aSsgw/pics/eI4an8aSsgw3114622.jpg'}, {'end': 3308.357, 'src': 'embed', 'start': 3256.732, 'weight': 4, 'content': [{'end': 3263.013, 'text': 'because the square root of one fourth x is the same thing as the square root of one fourth times the square root of x,', 'start': 3256.732, 'duration': 6.281}, {'end': 3266.834, 'text': 'which is the same thing as one half times the square root of x.', 'start': 3263.013, 'duration': 3.821}, {'end': 3277.112, 'text': 'And So now we can see algebraically that a vertical shrink by a factor of one half is the same as a horizontal stretch by a factor of four,', 'start': 3266.834, 'duration': 10.278}, {'end': 3279.213, 'text': 'at least for this function, the square root function.', 'start': 3277.112, 'duration': 2.101}, {'end': 3285.175, 'text': "This video gave some rules for transformations of functions, which I'll repeat below.", 'start': 3280.954, 'duration': 4.221}, {'end': 3291.217, 'text': 'Numbers on the outside correspond to changes in the y values.', 'start': 3286.335, 'duration': 4.882}, {'end': 3294.093, 'text': 'or vertical motions.', 'start': 3292.373, 'duration': 1.72}, {'end': 3301.575, 'text': 'Numbers on the inside of the function affect the x values and result in horizontal motions.', 'start': 3295.494, 'duration': 6.081}, {'end': 3308.357, 'text': 'Adding and subtracting corresponds to translations or shifts.', 'start': 3302.896, 'duration': 5.461}], 'summary': 'Square root of 1/4x equals 1/2 times square root of x. vertical shrink by 1/2 same as horizontal stretch by 4. rules for function transformations: outer numbers affect y values, inner numbers affect x values.', 'duration': 51.625, 'max_score': 3256.732, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/eI4an8aSsgw/pics/eI4an8aSsgw3256732.jpg'}, {'end': 3378.357, 'src': 'embed', 'start': 3352.011, 'weight': 6, 'content': [{'end': 3360.153, 'text': "a piecewise function is a function who's defined in pieces by two or more different rules that apply for different x values.", 'start': 3352.011, 'duration': 8.142}, {'end': 3370.515, 'text': 'The rule for calculating this function is you calculate negative x squared if x is less than one, but you calculate negative two, x plus three,', 'start': 3361.693, 'duration': 8.822}, {'end': 3372.236, 'text': 'if x is greater than or equal to one.', 'start': 3370.515, 'duration': 1.721}, {'end': 3378.357, 'text': 'So if we want to find f of negative two, well, negative two is less than one.', 'start': 3373.436, 'duration': 4.921}], 'summary': 'A piecewise function is defined by rules for different x values: f(-2)=-(-2)^2', 'duration': 26.346, 'max_score': 3352.011, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/eI4an8aSsgw/pics/eI4an8aSsgw3352011.jpg'}, {'end': 3558.417, 'src': 'embed', 'start': 3533.738, 'weight': 8, 'content': [{'end': 3539.982, 'text': "And in the first rule where we don't actually include that point, but we kind of have to draw it in order to anyway to draw the open circle here.", 'start': 3533.738, 'duration': 6.244}, {'end': 3544.086, 'text': 'Now the last question asks us if this function is continuous.', 'start': 3541.343, 'duration': 2.743}, {'end': 3551.792, 'text': "The informal definition of continuity that I'm going to use is that a function is continuous if you can draw the whole thing without picking up your pencil.", 'start': 3545.527, 'duration': 6.265}, {'end': 3558.417, 'text': "And in this case, we can't because we have to pick up our pencil to get from the jump here up to here.", 'start': 3552.352, 'duration': 6.065}], 'summary': 'The function is not continuous due to a jump.', 'duration': 24.679, 'max_score': 3533.738, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/eI4an8aSsgw/pics/eI4an8aSsgw3533738.jpg'}], 'start': 2618.722, 'title': 'Function transformations and piecewise functions', 'summary': 'Explains function transformations of the square root function, demonstrating how numbers inside and outside the function affect vertical and horizontal motions, and how multiplication, division, and negative signs result in stretching, shrinking, and reflection of the graph. it also covers four types of function transformations and introduces piecewise functions, explaining their definition and graphing using multiple rules, illustrated with an example and a discussion on function continuity at x = 1.', 'chapters': [{'end': 3137.284, 'start': 2618.722, 'title': 'Function transformations with square root', 'summary': 'Explains function transformations of the square root function, demonstrating how numbers inside and outside the function affect vertical and horizontal motions, and how multiplication, division, and negative signs result in stretching, shrinking, and reflection of the graph.', 'duration': 518.562, 'highlights': ['Numbers on the outside of the function affect the y values and result in vertical motions like subtracting 2 moves the graph down by 2, adding 2 moves the graph up by 2.', 'Numbers on the inside of the function affect the x values and result in a horizontal motion, going in the opposite direction, such as the minus 2 on the inside actually shifting the graph to the right.', 'Multiplication results in stretching or shrinking of the graph, such as y equals three times the square root of x stretches the graph vertically by a factor of three.', 'A negative sign results in a reflection, e.g., y equals the square root of negative x leads to a horizontal reflection, while y equals negative the square root of x results in a vertical reflection.']}, {'end': 3330.32, 'start': 3140.305, 'title': 'Function transformations and rules', 'summary': 'Explains four types of function transformations: vertical shifts, horizontal shifts, vertical stretches and reflections, and horizontal stretches and shrinks, with examples and algebraic explanations.', 'duration': 190.015, 'highlights': ['The video explains four types of function transformations: vertical shifts, horizontal shifts, vertical stretches and reflections, and horizontal stretches and shrinks.', 'Numbers on the outside affect the y values, resulting in vertical motions, while numbers on the inside affect the x values, resulting in horizontal motions.', 'The examples provided illustrate the impact of adding, subtracting, multiplying, and reflecting on the function graph.', 'Algebraic explanations reveal the relationship between vertical shrink by a factor of one half and horizontal stretch by a factor of four for the square root function.']}, {'end': 3574.035, 'start': 3331.58, 'title': 'Piecewise functions introduction', 'summary': "Introduces piecewise functions, explaining their definition and how to graph them using two or more rules, illustrated with an example of f(x) = -x^2 for x < 1 and -2x+3 for x >= 1, and concludes by discussing the function's continuity at x = 1.", 'duration': 242.455, 'highlights': ['Piecewise function definition and example', 'Graphing piecewise functions', 'Discussion on function continuity']}], 'duration': 955.313, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/eI4an8aSsgw/pics/eI4an8aSsgw2618722.jpg', 'highlights': ['Numbers on the outside affect the y values, resulting in vertical motions, e.g., subtracting 2 moves the graph down by 2.', 'Multiplication results in stretching or shrinking of the graph, e.g., y equals three times the square root of x stretches the graph vertically by a factor of three.', 'Numbers on the inside affect the x values, resulting in a horizontal motion, e.g., the minus 2 on the inside actually shifts the graph to the right.', 'A negative sign results in a reflection, e.g., y equals the square root of negative x leads to a horizontal reflection.', 'Explains four types of function transformations: vertical shifts, horizontal shifts, vertical stretches and reflections, and horizontal stretches and shrinks.', 'Algebraic explanations reveal the relationship between vertical shrink by a factor of one half and horizontal stretch by a factor of four for the square root function.', 'Piecewise function definition and example', 'Graphing piecewise functions', 'Discussion on function continuity']}, {'end': 4456.482, 'segs': [{'end': 3623.245, 'src': 'embed', 'start': 3574.696, 'weight': 0, 'content': [{'end': 3581.521, 'text': "However, it's possible to have a piecewise defined function that has no discontinuity if the two pieces happen to line up perfectly.", 'start': 3574.696, 'duration': 6.825}, {'end': 3587.406, 'text': "For example, if we've changed the function's definition slightly, I'll call it g of x's time.", 'start': 3583.363, 'duration': 4.043}, {'end': 3594.905, 'text': 'to still be the negative x squared when x is less than one, but this time negative two x plus one.', 'start': 3587.981, 'duration': 6.924}, {'end': 3601.73, 'text': 'if x is bigger than or equal to one, then when we graph it, the parabola piece will look the same.', 'start': 3594.905, 'duration': 6.825}, {'end': 3607.314, 'text': 'But the linear piece will be two units lower than before.', 'start': 3602.851, 'duration': 4.463}, {'end': 3614.098, 'text': "And so it'll actually start right here at one negative one and go down, and our function will be continuous.", 'start': 3607.814, 'duration': 6.284}, {'end': 3618.161, 'text': "That's all for this introduction to piecewise functions.", 'start': 3615.899, 'duration': 2.262}, {'end': 3623.245, 'text': 'The inverse of a function undoes what the function does.', 'start': 3620.322, 'duration': 2.923}], 'summary': 'Piecewise function can be continuous if pieces line up perfectly. introduction to piecewise functions and inverse functions.', 'duration': 48.549, 'max_score': 3574.696, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/eI4an8aSsgw/pics/eI4an8aSsgw3574696.jpg'}, {'end': 3783.53, 'src': 'embed', 'start': 3753.493, 'weight': 3, 'content': [{'end': 3755.835, 'text': 'Over the mirror line, y equals x.', 'start': 3753.493, 'duration': 2.342}, {'end': 3762.788, 'text': 'So our second key fact is that the graph of y equals f, inverse of x,', 'start': 3757.386, 'duration': 5.402}, {'end': 3770.13, 'text': 'can be obtained from the graph of y equals f of x by reflecting over the line y equals x.', 'start': 3762.788, 'duration': 7.342}, {'end': 3774.711, 'text': 'This makes sense because inverses reverse the roles of y and x.', 'start': 3770.13, 'duration': 4.581}, {'end': 3781.409, 'text': "In this same example, let's compute f inverse of f of 2.", 'start': 3776.567, 'duration': 4.842}, {'end': 3783.53, 'text': 'This open circle means composition.', 'start': 3781.409, 'duration': 2.121}], 'summary': 'The graph of y=f inverse of x is obtained by reflecting y=f(x) over line y=x.', 'duration': 30.037, 'max_score': 3753.493, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/eI4an8aSsgw/pics/eI4an8aSsgw3753493.jpg'}, {'end': 4378.257, 'src': 'embed', 'start': 4349.882, 'weight': 5, 'content': [{'end': 4352.124, 'text': 'If you look closely at these domains and ranges,', 'start': 4349.882, 'duration': 2.242}, {'end': 4362.269, 'text': "you'll notice that the domain of P corresponds exactly to the range of P inverse and the range of P corresponds to the domain of P inverse.", 'start': 4352.124, 'duration': 10.145}, {'end': 4370.053, 'text': 'This makes sense because inverse functions reverse the rules of y and x.', 'start': 4364.41, 'duration': 5.643}, {'end': 4378.257, 'text': 'The domain of f inverse of x is the x values for f inverse, which corresponds to the y values, or the range of f.', 'start': 4370.053, 'duration': 8.204}], 'summary': 'Inverse functions: domain of p is range of p inverse; range of p is domain of p inverse.', 'duration': 28.375, 'max_score': 4349.882, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/eI4an8aSsgw/pics/eI4an8aSsgw4349882.jpg'}, {'end': 4456.482, 'src': 'embed', 'start': 4413.362, 'weight': 4, 'content': [{'end': 4420.144, 'text': 'When we compose f with f inverse, we get the identity function y equals x.', 'start': 4413.362, 'duration': 6.782}, {'end': 4426.467, 'text': 'And similarly, when we compose f inverse with f, that brings x to x.', 'start': 4420.144, 'duration': 6.323}, {'end': 4428.908, 'text': 'In other words, f and f inverse undo each other.', 'start': 4426.467, 'duration': 2.441}, {'end': 4446.395, 'text': 'The function f of x has an inverse function if and only if the graph of y equals f of x satisfies the horizontal line test.', 'start': 4428.928, 'duration': 17.467}, {'end': 4456.482, 'text': 'And finally, the domain of f is the range of f inverse.', 'start': 4448.316, 'duration': 8.166}], 'summary': 'F and f inverse undo each other, satisfy horizontal line test, and have corresponding domains and ranges.', 'duration': 43.12, 'max_score': 4413.362, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/eI4an8aSsgw/pics/eI4an8aSsgw4413362.jpg'}], 'start': 3574.696, 'title': 'Piecewise functions and inverse functions', 'summary': 'Explains the continuity of piecewise functions without discontinuities using an example, and introduces inverse functions, discussing their properties, graph symmetry, compositions, systematic methods for finding inverses, and key properties.', 'chapters': [{'end': 3623.245, 'start': 3574.696, 'title': 'Piecewise functions and continuity', 'summary': 'Explains how a piecewise function can be continuous, even without discontinuities, by aligning the two pieces perfectly, as demonstrated by the example of a function being continuous when x is less than one, and then shifting down by two units when x is bigger than or equal to one.', 'duration': 48.549, 'highlights': ["The function g of x's time can be continuous without any discontinuity if the two pieces line up perfectly.", 'The linear piece will be two units lower than before, starting at one negative one and going down, resulting in a continuous function.', 'Understanding that the inverse of a function undoes what the function does is important in the context of piecewise functions.']}, {'end': 4456.482, 'start': 3623.865, 'title': 'Introduction to inverse functions', 'summary': 'Introduces inverse functions and their properties, discussing the relationship between inverse functions, graph symmetry, compositions of functions, systematic methods for finding inverses, examples of functions with and without inverses, and the key properties of inverse functions.', 'duration': 832.617, 'highlights': ['Inverse functions reverse the roles of y and x.', 'The graph of y equals f inverse of x is obtained from the graph of y equals f of x by reflecting over the line y equals x.', 'The function f of x has an inverse function if and only if the graph of y equals f of x satisfies the horizontal line test.', 'The domain of f is the range of f inverse.', 'Compositions of f with f inverse and f inverse with f result in the identity function y equals x.']}], 'duration': 881.786, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/eI4an8aSsgw/pics/eI4an8aSsgw3574696.jpg', 'highlights': ["The function g of x's time can be continuous without any discontinuity if the two pieces line up perfectly.", 'The linear piece will be two units lower than before, starting at one negative one and going down, resulting in a continuous function.', 'Understanding that the inverse of a function undoes what the function does is important in the context of piecewise functions.', 'The graph of y equals f inverse of x is obtained from the graph of y equals f of x by reflecting over the line y equals x.', 'The function f of x has an inverse function if and only if the graph of y equals f of x satisfies the horizontal line test.', 'The domain of f is the range of f inverse.', 'Compositions of f with f inverse and f inverse with f result in the identity function y equals x.']}, {'end': 6495.977, 'segs': [{'end': 4558.499, 'src': 'embed', 'start': 4521.345, 'weight': 0, 'content': [{'end': 4527.27, 'text': "The 2 pi was chosen because that's the circumference of a circle with the radius of 1.", 'start': 4521.345, 'duration': 5.925}, {'end': 4533.755, 'text': 'A half circle is then half of 2 pi radians, which is pi radians.', 'start': 4527.27, 'duration': 6.485}, {'end': 4543.904, 'text': 'And if I wanted to do, say, a quarter of a circle, that would be a quarter of my 2 pi radians.', 'start': 4535.176, 'duration': 8.728}, {'end': 4549.895, 'text': 'A quarter of 2 pi is pi over 2.', 'start': 4544.344, 'duration': 5.551}, {'end': 4558.499, 'text': "To convert between degrees and radians, it's handy to use the fact that 180 degrees corresponds to pi radians.", 'start': 4549.895, 'duration': 8.604}], 'summary': 'Using radians, a quarter of a circle is pi/2 radians.', 'duration': 37.154, 'max_score': 4521.345, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/eI4an8aSsgw/pics/eI4an8aSsgw4521345.jpg'}, {'end': 5617.174, 'src': 'heatmap', 'start': 4829.775, 'weight': 0.753, 'content': [{'end': 4844.326, 'text': 'Okay, now I can write this as 32 degrees plus 17 over 60 degrees plus 25 over 3600 degrees because the minutes cancelled and the seconds cancelled.', 'start': 4829.775, 'duration': 14.551}, {'end': 4853.153, 'text': 'If I add this all up on my calculator, I get 32.2903 degrees up to four decimal places.', 'start': 4845.627, 'duration': 7.526}, {'end': 4861.035, 'text': 'I can also go the other direction and convert a decimal number of degrees into degrees, minutes and seconds.', 'start': 4854.434, 'duration': 6.601}, {'end': 4870.622, 'text': "I'm starting out with 247 degrees plus a decimal number of degrees left over.", 'start': 4862.416, 'duration': 8.206}, {'end': 4878.248, 'text': 'If I want to convert 0.3486 degrees to minutes.', 'start': 4872.023, 'duration': 6.225}, {'end': 4882.371, 'text': 'well, I know that I have 60 minutes in one degree.', 'start': 4878.248, 'duration': 4.123}, {'end': 4889.186, 'text': 'this time I want the minutes on the top and the degrees on the bottom, so that my degrees will cancel.', 'start': 4882.371, 'duration': 6.815}, {'end': 4892.848, 'text': 'So I just multiply my decimal by 60.', 'start': 4890.166, 'duration': 2.682}, {'end': 4897.972, 'text': 'And I end up with 20.916 minutes.', 'start': 4892.848, 'duration': 5.124}, {'end': 4900.133, 'text': 'Let me copy this down.', 'start': 4898.892, 'duration': 1.241}, {'end': 4907.899, 'text': "And now I'm going to take the 0.916 minutes and convert those to seconds.", 'start': 4901.794, 'duration': 6.105}, {'end': 4912.942, 'text': "So I know that they're 60 seconds in one minute.", 'start': 4908.919, 'duration': 4.023}, {'end': 4924.452, 'text': 'So canceling my units again, I just multiply by 60 which gives me 54.96 minutes.', 'start': 4913.683, 'duration': 10.769}, {'end': 4936.055, 'text': 'I can copy this down as is or I can round off my seconds and get 247 degrees 20 minutes and 55 seconds.', 'start': 4925.533, 'duration': 10.522}, {'end': 4942.457, 'text': 'Converting between different ways of measuring angles is all about unit conversion.', 'start': 4938.916, 'duration': 3.541}, {'end': 4951.285, 'text': 'To convert between radians and degrees, we use the fact that pi radians corresponds to 180 degrees.', 'start': 4943.783, 'duration': 7.502}, {'end': 4965.729, 'text': 'To convert between minutes and seconds and degrees, we use the fact that 60 minutes corresponds to 1 degree and 60 seconds corresponds to 1 minute.', 'start': 4953.046, 'duration': 12.683}, {'end': 4973.031, 'text': 'If you have a circle, a piece of the circumference is called an arc.', 'start': 4967.029, 'duration': 6.002}, {'end': 4979.969, 'text': 'and a wedge of pi for that circle is called a sector.', 'start': 4974.785, 'duration': 5.184}, {'end': 4991.378, 'text': 'This video explains how to calculate the length of the arc and the area of the sector in terms of the angle and the radius of the circle.', 'start': 4981.851, 'duration': 9.527}, {'end': 5000.645, 'text': 'The circumference of a circle is given by the formula, the circumference equals two pi times the radius.', 'start': 4993.559, 'duration': 7.086}, {'end': 5004.127, 'text': "Let's use that to solve the following problem.", 'start': 5002.045, 'duration': 2.082}, {'end': 5008.67, 'text': 'A circular pool has a radius of 8 meters.', 'start': 5005.047, 'duration': 3.623}, {'end': 5014.795, 'text': 'Find the arc length spanned by a central angle of 2.5 radians.', 'start': 5010.191, 'duration': 4.604}, {'end': 5018.678, 'text': 'So this angle from the center of the circle is supposed to be 2.5 radians.', 'start': 5015.455, 'duration': 3.223}, {'end': 5029.036, 'text': "I've drawn the angle is a little bit less than half the circle because half the circle would be pi radians, which is 3.141 radians.", 'start': 5021.889, 'duration': 7.147}, {'end': 5039.707, 'text': 'So I want to find the length of this arc, I know that the total circumference is going to be two pi times eight meters.', 'start': 5029.937, 'duration': 9.77}, {'end': 5043.21, 'text': 'But I just want a fraction of the circumference.', 'start': 5041.148, 'duration': 2.062}, {'end': 5055.298, 'text': 'The arc length I want is going to be given by the fraction of the circle that the angle makes times the circumference of the circle.', 'start': 5044.632, 'duration': 10.666}, {'end': 5067.72, 'text': 'The fraction of the circle that I want is given by the ratio of 2.5 radians over the total number of radians in the circle, which is 2 pi radians.', 'start': 5057.298, 'duration': 10.422}, {'end': 5073.061, 'text': 'Multiply that by the circumference, which we said was 2 pi times 8 meters.', 'start': 5068.96, 'duration': 4.101}, {'end': 5086.95, 'text': "Notice that the two pi's cancel and I'm left with just 2.5 times eight meters, which is 10 meters.", 'start': 5075.868, 'duration': 11.082}, {'end': 5089.891, 'text': 'In general,', 'start': 5088.511, 'duration': 1.38}, {'end': 5100.793, 'text': 'the arc length is related to the angle it spans by considering the arc length to be the fraction of the circle you get by taking the angle over the total angle,', 'start': 5089.891, 'duration': 10.902}, {'end': 5105.654, 'text': 'the circle of two pi times the circumference, which is two pi r.', 'start': 5100.793, 'duration': 4.861}, {'end': 5114.155, 'text': "Since the two pi's cancel, as they did in our example, that means that arc length is given by theta times r.", 'start': 5107.168, 'duration': 6.987}, {'end': 5120.652, 'text': 'Here r is the radius of the circle, and theta is the angle.', 'start': 5115.99, 'duration': 4.662}, {'end': 5127.955, 'text': "It's important to note that theta needs to be measured in radians, not degrees, for this to work.", 'start': 5121.592, 'duration': 6.363}, {'end': 5138.099, 'text': "That's because when I took this ratio, I was using 2 pi radians for the total measure of the circle, so it's important that theta also be in radians.", 'start': 5128.835, 'duration': 9.264}, {'end': 5144.041, 'text': 'The area of a circle is given by the formula pi r squared.', 'start': 5139.519, 'duration': 4.522}, {'end': 5145.94, 'text': 'where r is the radius.', 'start': 5144.639, 'duration': 1.301}, {'end': 5157.688, 'text': "Let's use that fact to find the area of the sector of a circle of radius 10 meters that spans an angle of pi over 6 radians.", 'start': 5147.441, 'duration': 10.247}, {'end': 5169.976, 'text': 'We know that the total area of the circle is given by pi times 10 meters squared, or 100 pi meters squared.', 'start': 5159.249, 'duration': 10.727}, {'end': 5176.535, 'text': 'But we just want a fraction of that area to give us the area of our sector.', 'start': 5171.277, 'duration': 5.258}, {'end': 5187.818, 'text': 'So we want to take the fraction of the circle that the sector makes times the area of the circle.', 'start': 5178.015, 'duration': 9.803}, {'end': 5198.042, 'text': 'Well, the fraction of the circle is given by the angle of pi over 6 over the total angle in the whole circle of 2 pi,', 'start': 5189.439, 'duration': 8.603}, {'end': 5204.358, 'text': "And we'll times that by the 100 pi meters squared.", 'start': 5200.916, 'duration': 3.442}, {'end': 5217.104, 'text': 'That simplifies to 100 pi over 12 meters squared or 25 thirds pi meters squared.', 'start': 5205.218, 'duration': 11.886}, {'end': 5223.687, 'text': "As a decimal, that's 26.18 meters squared up to two decimal places.", 'start': 5218.064, 'duration': 5.623}, {'end': 5227.269, 'text': 'In general, if you want to find the area of a sector,', 'start': 5224.728, 'duration': 2.541}, {'end': 5240.009, 'text': 'you can take the fraction of the circle that the sector spans out times the area of the circle pi r squared.', 'start': 5230.526, 'duration': 9.483}, {'end': 5251.133, 'text': 'Notice that the fraction of the circle is given by the angle that the sector makes in radians divided by the two pi radians in the circle.', 'start': 5241.11, 'duration': 10.023}, {'end': 5256.335, 'text': "Since the pi's cancel.", 'start': 5254.514, 'duration': 1.821}, {'end': 5270.617, 'text': 'we can simplify this formula by saying the area is given by theta over 2 times r squared, where theta is the angle of the sector in radians.', 'start': 5256.335, 'duration': 14.282}, {'end': 5279.179, 'text': "Again, it's important that this angle be in radians, since that's how we're doing our fraction of our circle comparison.", 'start': 5272.498, 'duration': 6.681}, {'end': 5283.5, 'text': 'As usual, r represents the radius of the circle.', 'start': 5280.459, 'duration': 3.041}, {'end': 5290.477, 'text': "Notice that if you don't have your angle in radians, if it's in degrees instead, not a big deal,", 'start': 5284.915, 'duration': 5.562}, {'end': 5294.579, 'text': 'because we can always convert from degrees to radians first before using our formula.', 'start': 5290.477, 'duration': 4.102}, {'end': 5309.985, 'text': 'In this video, we saw that for a sector of angle theta given in radians, the arc length is given by the formula theta times r.', 'start': 5295.899, 'duration': 14.086}, {'end': 5319.315, 'text': 'and the area of the sector is given by theta over two times r squared, where r is the radius of the circle.', 'start': 5310.891, 'duration': 8.424}, {'end': 5328.379, 'text': 'This video defines linear and radial speed for a rotating circle and explains how to convert between them.', 'start': 5320.456, 'duration': 7.923}, {'end': 5331.08, 'text': 'Consider a spinning wheel.', 'start': 5329.82, 'duration': 1.26}, {'end': 5337.704, 'text': 'The angular speed is the angle it goes through in a unit of time.', 'start': 5331.861, 'duration': 5.843}, {'end': 5343.753, 'text': 'I can write that as angle per time.', 'start': 5339.444, 'duration': 4.309}, {'end': 5352.455, 'text': 'The units will be something like radians per second, if the angles measured in radians and the times measured in seconds,', 'start': 5345.073, 'duration': 7.382}, {'end': 5355.455, 'text': 'or I guess we could do degrees per minute, etc.', 'start': 5352.455, 'duration': 3}, {'end': 5362.217, 'text': 'The linear speed is the speed of a point on the rim of the wheel.', 'start': 5357.536, 'duration': 4.681}, {'end': 5374.958, 'text': 'So that is the distance a point on the rim of the wheel travels in a unit of time.', 'start': 5363.037, 'duration': 11.921}, {'end': 5380.939, 'text': 'I can think of that as distance per time.', 'start': 5377.778, 'duration': 3.161}, {'end': 5389.321, 'text': 'And so it has units of something like meters per second, for example, or could be feet per minute, etc.', 'start': 5381.499, 'duration': 7.822}, {'end': 5394.762, 'text': 'Suppose we have a Ferris wheel with radius 20 meters.', 'start': 5390.761, 'duration': 4.001}, {'end': 5400.263, 'text': "That's making one revolution every two minutes.", 'start': 5397.382, 'duration': 2.881}, {'end': 5405.915, 'text': 'We want to find its angular speed and the linear speed of a point on its rim.', 'start': 5401.854, 'duration': 4.061}, {'end': 5412.318, 'text': 'Now the Ferris wheel is going one revolution every two minutes.', 'start': 5406.616, 'duration': 5.702}, {'end': 5416.439, 'text': "So that's one half revolution per minute.", 'start': 5412.538, 'duration': 3.901}, {'end': 5422.601, 'text': 'I want to find the angular speed, which is the angle it goes through in a unit of time.', 'start': 5416.739, 'duration': 5.862}, {'end': 5430.304, 'text': "I've already got a unit of time in the denominator minutes, but I've got to somehow convert revolutions to angle.", 'start': 5424.262, 'duration': 6.042}, {'end': 5442.945, 'text': "So my angular speed, one half revolutions per minute, one revolution, I'll put that on the bottom, so the units will cancel out.", 'start': 5431.698, 'duration': 11.247}, {'end': 5447.848, 'text': 'One revolution is going through an angle of two pi radians.', 'start': 5443.825, 'duration': 4.023}, {'end': 5459.555, 'text': 'That means my angular speed is going to be one half times two pi radians per minute, or pi radians per minute.', 'start': 5449.889, 'duration': 9.666}, {'end': 5476.9, 'text': 'the linear speed, the speed of a point on the rim, well, the wheel is going pi radians per minute, I got to somehow convert radians into distance.', 'start': 5462.932, 'duration': 13.968}, {'end': 5487.766, 'text': "So I know that when I go to pi radians, so that's all the way around the circle, I'm going all the way around the circumference and distance.", 'start': 5477.6, 'duration': 10.166}, {'end': 5492.412, 'text': 'So that would be two pi times the radius or two pi times 20.', 'start': 5488.066, 'duration': 4.346}, {'end': 5504.618, 'text': "meters The two pi's cancel, as do the radians, and I'm left with 20 pi meters per minute.", 'start': 5492.412, 'duration': 12.206}, {'end': 5511.041, 'text': "Let's review how we got from angular speed to linear speed in order to get a more general formula.", 'start': 5505.959, 'duration': 5.082}, {'end': 5513.983, 'text': "I'll call the angular speed.", 'start': 5512.622, 'duration': 1.361}, {'end': 5519.945, 'text': "Omega looks like a W and I'll call the linear speed V.", 'start': 5514.963, 'duration': 4.982}, {'end': 5532.569, 'text': 'In the previous problem, We found the linear speed v by starting with the angular speed omega and multiplying it by the circumference divided by 2 pi.', 'start': 5519.945, 'duration': 12.624}, {'end': 5543.017, 'text': "That's because the point on the rim travels the whole circumference as it goes through an angle of 2 pi all the way around the circle.", 'start': 5533.47, 'duration': 9.547}, {'end': 5548.181, 'text': "So I'll write that down here, circumference over 2 pi.", 'start': 5543.858, 'duration': 4.323}, {'end': 5561.516, 'text': "But since I know that my circumference is given by the formula 2, pi, r and the 2 pi's cancel as they did in the previous problem.", 'start': 5551.311, 'duration': 10.205}, {'end': 5568.98, 'text': 'that shows that the linear speed is the angular speed times r, where r is the radius of the circle.', 'start': 5561.516, 'duration': 7.464}, {'end': 5581.066, 'text': 'In this video, we defined linear speed and radial speed and found that linear speed was radial speed times the radius of the circle.', 'start': 5572.442, 'duration': 8.624}, {'end': 5596.337, 'text': 'This video introduces the trig functions sine, cosine, tangent, secant, cosecant, and cotangent for right triangles.', 'start': 5582.728, 'duration': 13.609}, {'end': 5617.174, 'text': 'For a right triangle with sides of length a, b, and c and an angle theta as drawn, we as the length of the opposite side over the hypotenuse.', 'start': 5600.38, 'duration': 16.794}], 'summary': 'The transcript covers angle conversions, arc length, area of sector, and linear speed calculations in radians and degrees.', 'duration': 787.399, 'max_score': 4829.775, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/eI4an8aSsgw/pics/eI4an8aSsgw4829775.jpg'}, {'end': 5138.099, 'src': 'embed', 'start': 5107.168, 'weight': 2, 'content': [{'end': 5114.155, 'text': "Since the two pi's cancel, as they did in our example, that means that arc length is given by theta times r.", 'start': 5107.168, 'duration': 6.987}, {'end': 5120.652, 'text': 'Here r is the radius of the circle, and theta is the angle.', 'start': 5115.99, 'duration': 4.662}, {'end': 5127.955, 'text': "It's important to note that theta needs to be measured in radians, not degrees, for this to work.", 'start': 5121.592, 'duration': 6.363}, {'end': 5138.099, 'text': "That's because when I took this ratio, I was using 2 pi radians for the total measure of the circle, so it's important that theta also be in radians.", 'start': 5128.835, 'duration': 9.264}], 'summary': 'Arc length is given by theta times r; theta must be in radians.', 'duration': 30.931, 'max_score': 5107.168, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/eI4an8aSsgw/pics/eI4an8aSsgw5107168.jpg'}, {'end': 5283.5, 'src': 'embed', 'start': 5256.335, 'weight': 1, 'content': [{'end': 5270.617, 'text': 'we can simplify this formula by saying the area is given by theta over 2 times r squared, where theta is the angle of the sector in radians.', 'start': 5256.335, 'duration': 14.282}, {'end': 5279.179, 'text': "Again, it's important that this angle be in radians, since that's how we're doing our fraction of our circle comparison.", 'start': 5272.498, 'duration': 6.681}, {'end': 5283.5, 'text': 'As usual, r represents the radius of the circle.', 'start': 5280.459, 'duration': 3.041}], 'summary': 'Area of sector = θ/2 * r², θ in radians', 'duration': 27.165, 'max_score': 5256.335, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/eI4an8aSsgw/pics/eI4an8aSsgw5256335.jpg'}, {'end': 5343.753, 'src': 'embed', 'start': 5310.891, 'weight': 3, 'content': [{'end': 5319.315, 'text': 'and the area of the sector is given by theta over two times r squared, where r is the radius of the circle.', 'start': 5310.891, 'duration': 8.424}, {'end': 5328.379, 'text': 'This video defines linear and radial speed for a rotating circle and explains how to convert between them.', 'start': 5320.456, 'duration': 7.923}, {'end': 5331.08, 'text': 'Consider a spinning wheel.', 'start': 5329.82, 'duration': 1.26}, {'end': 5337.704, 'text': 'The angular speed is the angle it goes through in a unit of time.', 'start': 5331.861, 'duration': 5.843}, {'end': 5343.753, 'text': 'I can write that as angle per time.', 'start': 5339.444, 'duration': 4.309}], 'summary': 'Explains linear and radial speed conversion for rotating circle.', 'duration': 32.862, 'max_score': 5310.891, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/eI4an8aSsgw/pics/eI4an8aSsgw5310891.jpg'}, {'end': 5416.439, 'src': 'embed', 'start': 5381.499, 'weight': 5, 'content': [{'end': 5389.321, 'text': 'And so it has units of something like meters per second, for example, or could be feet per minute, etc.', 'start': 5381.499, 'duration': 7.822}, {'end': 5394.762, 'text': 'Suppose we have a Ferris wheel with radius 20 meters.', 'start': 5390.761, 'duration': 4.001}, {'end': 5400.263, 'text': "That's making one revolution every two minutes.", 'start': 5397.382, 'duration': 2.881}, {'end': 5405.915, 'text': 'We want to find its angular speed and the linear speed of a point on its rim.', 'start': 5401.854, 'duration': 4.061}, {'end': 5412.318, 'text': 'Now the Ferris wheel is going one revolution every two minutes.', 'start': 5406.616, 'duration': 5.702}, {'end': 5416.439, 'text': "So that's one half revolution per minute.", 'start': 5412.538, 'duration': 3.901}], 'summary': 'Ferris wheel with 20m radius rotates at 0.5 revolutions/minute.', 'duration': 34.94, 'max_score': 5381.499, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/eI4an8aSsgw/pics/eI4an8aSsgw5381499.jpg'}, {'end': 5617.174, 'src': 'embed', 'start': 5572.442, 'weight': 9, 'content': [{'end': 5581.066, 'text': 'In this video, we defined linear speed and radial speed and found that linear speed was radial speed times the radius of the circle.', 'start': 5572.442, 'duration': 8.624}, {'end': 5596.337, 'text': 'This video introduces the trig functions sine, cosine, tangent, secant, cosecant, and cotangent for right triangles.', 'start': 5582.728, 'duration': 13.609}, {'end': 5617.174, 'text': 'For a right triangle with sides of length a, b, and c and an angle theta as drawn, we as the length of the opposite side over the hypotenuse.', 'start': 5600.38, 'duration': 16.794}], 'summary': 'Introduction of trig functions for right triangles and their application in finding the length of the opposite side over the hypotenuse.', 'duration': 44.732, 'max_score': 5572.442, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/eI4an8aSsgw/pics/eI4an8aSsgw5572442.jpg'}, {'end': 5970.101, 'src': 'embed', 'start': 5938.88, 'weight': 6, 'content': [{'end': 5946.503, 'text': "So if we have a kite that's flying at an angle of elevation, that's the angle from the horizontal of 75 degrees.", 'start': 5938.88, 'duration': 7.623}, {'end': 5952.509, 'text': 'With a kite length string of 100 meters.', 'start': 5949.107, 'duration': 3.402}, {'end': 5954.391, 'text': 'we want to find out how high it is.', 'start': 5952.509, 'duration': 1.882}, {'end': 5957.072, 'text': "I'll call the height y.", 'start': 5954.651, 'duration': 2.421}, {'end': 5966.419, 'text': 'Well, we want to relate the known quantities, this angle and this hypotenuse, to the unknown quantity.', 'start': 5957.072, 'duration': 9.347}, {'end': 5970.101, 'text': 'The unknown quantity is the opposite side of our triangle.', 'start': 5967.299, 'duration': 2.802}], 'summary': 'A kite flying at 75 degrees elevation with a 100-meter string, aiming to find its height.', 'duration': 31.221, 'max_score': 5938.88, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/eI4an8aSsgw/pics/eI4an8aSsgw5938880.jpg'}, {'end': 6281.376, 'src': 'embed', 'start': 6254.437, 'weight': 7, 'content': [{'end': 6261.562, 'text': 'And a similar computation shows that cosine of 45 degrees is also square root of two over two as before.', 'start': 6254.437, 'duration': 7.125}, {'end': 6266.586, 'text': 'This makes sense because sine and cosine are based on ratios of sides.', 'start': 6262.463, 'duration': 4.123}, {'end': 6272.61, 'text': "And since these two triangles are similar triangles, they'll have the same ratios of sides.", 'start': 6267.166, 'duration': 5.444}, {'end': 6281.376, 'text': "To find the sine and cosine of 30 degrees, let's use this 30 6090 right triangle with hypotenuse one.", 'start': 6274.271, 'duration': 7.105}], 'summary': 'Sine and cosine ratios in similar triangles are consistent, e.g., cos 45° = √2/2.', 'duration': 26.939, 'max_score': 6254.437, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/eI4an8aSsgw/pics/eI4an8aSsgw6254437.jpg'}, {'end': 6397.267, 'src': 'embed', 'start': 6371.414, 'weight': 8, 'content': [{'end': 6379.956, 'text': 'So we get a sine of one half over one which is one half cosine of 30 degrees is adjacent over hypotenuse.', 'start': 6371.414, 'duration': 8.542}, {'end': 6386.065, 'text': "So that's the square root of 3 over 2 divided by 1.", 'start': 6381.604, 'duration': 4.461}, {'end': 6389.405, 'text': 'To find sine of 60 degrees and cosine of 60 degrees.', 'start': 6386.065, 'duration': 3.34}, {'end': 6397.267, 'text': 'we can actually use this same green triangle and just focus on this upper corner angle of 60 degrees instead.', 'start': 6389.405, 'duration': 7.862}], 'summary': 'Calculating sine and cosine of 30 and 60 degrees using trigonometric ratios.', 'duration': 25.853, 'max_score': 6371.414, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/eI4an8aSsgw/pics/eI4an8aSsgw6371414.jpg'}], 'start': 4457.783, 'title': 'Angle conversions and trigonometry', 'summary': 'Covers the conversion between degrees and radians, angles between degrees, minutes, and seconds to decimal degrees, conversion of radians and degrees, calculation of arc length and sector area, and angular and linear speed in rotating circles, along with trigonometric functions for right triangles and triangle computations.', 'chapters': [{'end': 4718.609, 'start': 4457.783, 'title': 'Converting between degrees and radians', 'summary': 'Explains the conversion between degrees and radians, including the relationships between them, and provides examples of converting angles from degrees to radians and vice versa, demonstrating the use of the conversion formula and providing numerical results.', 'duration': 260.826, 'highlights': ['Converting seven radians to degrees', 'Converting negative 135 degrees to radians', 'The relationship between degrees and radians']}, {'end': 4942.457, 'start': 4719.99, 'title': 'Angle conversion and unit measurement', 'summary': 'Explains the conversion of angles between degrees, minutes, and seconds to decimal degrees and vice versa, including the relevant conversion factors, such as 60 minutes in one degree and 3600 seconds in one degree.', 'duration': 222.467, 'highlights': ['The conversion of 32 degrees, 17 minutes, and 25 seconds to a decimal number of degrees results in 32.2903 degrees.', 'The process of converting 0.3486 degrees to 20 minutes and 55 seconds involves multiplying the decimal by 60 to get the minutes and then multiplying the remaining decimal by 60 to obtain the seconds.', 'The concept of unit conversion is emphasized as an integral aspect of converting between different ways of measuring angles.']}, {'end': 5309.985, 'start': 4943.783, 'title': 'Radian-degree conversion & circle calculations', 'summary': 'Explains how to convert between radians and degrees, calculate the length of the arc and the area of the sector in terms of the angle and the radius, and uses an example of a circular pool with a radius of 8 meters and a central angle of 2.5 radians to demonstrate the calculations.', 'duration': 366.202, 'highlights': ['The circumference of a circle is given by the formula, the circumference equals two pi times the radius.', 'The arc length is given by the formula theta times r, where theta is the angle in radians and r is the radius of the circle.', 'The area of a sector is given by the formula theta over 2 times r squared, where theta is the angle of the sector in radians.']}, {'end': 5568.98, 'start': 5310.891, 'title': 'Angular and linear speed of rotating circle', 'summary': 'Explains the concept of angular and linear speed in a rotating circle, with an example of finding the angular and linear speed of a ferris wheel with a radius of 20 meters making one revolution every two minutes.', 'duration': 258.089, 'highlights': ['The angular speed of the Ferris wheel is pi radians per minute, which is derived from the given information about its revolution per minute and the conversion of revolutions to angle.', 'The formula for converting angular speed to linear speed is demonstrated as linear speed = angular speed * radius, providing a more general formula for the relationship between angular and linear speed.', 'The detailed process of converting angular speed to linear speed is illustrated using the example of the Ferris wheel, where the linear speed is determined to be 20 pi meters per minute.', 'The relationship between angular speed, linear speed, and the radius of the circle is emphasized, showing that the linear speed is directly proportional to the angular speed and the radius of the circle.']}, {'end': 5937.96, 'start': 5572.442, 'title': 'Trigonometric functions and right triangles', 'summary': 'Introduces trigonometric functions sine, cosine, and tangent for right triangles, along with their relationships and applications, before determining the exact values of the six trigonometric functions for a given angle in a triangle.', 'duration': 365.518, 'highlights': ['The chapter introduces trigonometric functions sine, cosine, and tangent for right triangles, along with their relationships and applications, before determining the exact values of the six trigonometric functions for a given angle in a triangle.', 'The relationship between tangent and sine and cosine is discussed, with tangent of theta equal to sine of theta over cosine of theta.', 'The exact values of all six trigonometric functions for the angle theta in a triangle are computed, including sine, cosine, tangent, secant, cosecant, and cotangent.', "The mnemonic 'SOH CAH TOA' is introduced to remember that sine is opposite over hypotenuse, cosine is adjacent over hypotenuse, and tangent is opposite over adjacent."]}, {'end': 6495.977, 'start': 5938.88, 'title': 'Trigonometry and triangle computation', 'summary': 'Explains how to calculate the height of a kite at a 75-degree angle, computes the sine and cosine of 45 and 30 degree angles using geometry, and provides a summary table of results.', 'duration': 557.097, 'highlights': ['The height of the kite at a 75-degree angle is 96.59 meters, calculated using the sine function and a string length of 100 meters.', 'The computation of the sine and cosine of a 45-degree angle using right triangles reveals that both are equal to the square root of 2 over 2.', 'The computation of the sine and cosine of a 30-degree angle using a right triangle yields the results of 1/2 and the square root of 3 over 2, respectively.']}], 'duration': 2038.194, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/eI4an8aSsgw/pics/eI4an8aSsgw4457783.jpg', 'highlights': ['The circumference of a circle is given by the formula, the circumference equals two pi times the radius.', 'The area of a sector is given by the formula theta over 2 times r squared, where theta is the angle of the sector in radians.', 'The arc length is given by the formula theta times r, where theta is the angle in radians and r is the radius of the circle.', 'The relationship between angular speed, linear speed, and the radius of the circle is emphasized, showing that the linear speed is directly proportional to the angular speed and the radius of the circle.', 'The formula for converting angular speed to linear speed is demonstrated as linear speed = angular speed * radius, providing a more general formula for the relationship between angular and linear speed.', 'The detailed process of converting angular speed to linear speed is illustrated using the example of the Ferris wheel, where the linear speed is determined to be 20 pi meters per minute.', 'The height of the kite at a 75-degree angle is 96.59 meters, calculated using the sine function and a string length of 100 meters.', 'The computation of the sine and cosine of a 45-degree angle using right triangles reveals that both are equal to the square root of 2 over 2.', 'The computation of the sine and cosine of a 30-degree angle using a right triangle yields the results of 1/2 and the square root of 3 over 2, respectively.', 'The chapter introduces trigonometric functions sine, cosine, and tangent for right triangles, along with their relationships and applications, before determining the exact values of the six trigonometric functions for a given angle in a triangle.']}, {'end': 7477.144, 'segs': [{'end': 6563.341, 'src': 'embed', 'start': 6499.24, 'weight': 0, 'content': [{'end': 6501.862, 'text': 'Doing a visual check you can easily fill in the angles.', 'start': 6499.24, 'duration': 2.622}, {'end': 6507.567, 'text': 'The smaller angle must be the 30 degree angle and the larger one must be the 60th degree.', 'start': 6502.523, 'duration': 5.044}, {'end': 6519.298, 'text': 'In this video we computed the sine and cosine of three special angles 30 degrees, 45 degrees and 60 degrees.', 'start': 6509.113, 'duration': 10.185}, {'end': 6526.322, 'text': 'This video defines sine and cosine in terms of points on the unit circle.', 'start': 6521.88, 'duration': 4.442}, {'end': 6530.084, 'text': 'A unit circle is a circle with radius one.', 'start': 6527.082, 'duration': 3.002}, {'end': 6536.507, 'text': 'Up to now, we define sine and cosine and tangent in terms of right triangles.', 'start': 6532.105, 'duration': 4.402}, {'end': 6539.428, 'text': 'For example, to find sine of 14 degrees.', 'start': 6537.047, 'duration': 2.381}, {'end': 6540.428, 'text': 'in theory,', 'start': 6539.428, 'duration': 1}, {'end': 6555.735, 'text': 'you could draw a right triangle with an angle of 14 degrees and then calculate the sine as the length of the opposite side over the length of hypotenuse.', 'start': 6540.428, 'duration': 15.307}, {'end': 6563.341, 'text': 'But if we use this method to try to compute sine of 120 degrees, things go horribly wrong.', 'start': 6557.795, 'duration': 5.546}], 'summary': 'Video discusses computing sine and cosine of 30, 45, and 60 degrees using the unit circle.', 'duration': 64.101, 'max_score': 6499.24, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/eI4an8aSsgw/pics/eI4an8aSsgw6499240.jpg'}, {'end': 6770.864, 'src': 'heatmap', 'start': 6566.244, 'weight': 0.771, 'content': [{'end': 6572.952, 'text': "When we draw this 120 degree angle, and this right angle, there's no way to complete this picture to get a right triangle.", 'start': 6566.244, 'duration': 6.708}, {'end': 6579.08, 'text': "So instead, we're going to use a unit circle, that is a circle of radius one.", 'start': 6574.054, 'duration': 5.026}, {'end': 6584.568, 'text': 'The figure below illustrates how right triangles and the unit circle are related.', 'start': 6580.285, 'duration': 4.283}, {'end': 6595.677, 'text': 'If you draw right triangles with larger of hypotenuse one with larger and larger angles, then the top vertex sweeps out part of a unit circle.', 'start': 6585.309, 'duration': 10.368}, {'end': 6599.32, 'text': "Let's look at this relationship in more detail.", 'start': 6597.198, 'duration': 2.122}, {'end': 6605.765, 'text': "In this figure, I've drawn a right triangle inside a unit circle.", 'start': 6600.921, 'duration': 4.844}, {'end': 6611.2, 'text': 'The hypotenuse of the triangle is the radius of the circle, which is one.', 'start': 6607.298, 'duration': 3.902}, {'end': 6615.802, 'text': 'One vertex of the right triangle is at the origin.', 'start': 6613.101, 'duration': 2.701}, {'end': 6621.005, 'text': 'Another vertex of the right triangle is at the edge of the circle.', 'start': 6615.822, 'duration': 5.183}, {'end': 6625.787, 'text': "I'm going to call the coordinates of that vertex AB.", 'start': 6622.025, 'duration': 3.762}, {'end': 6638.296, 'text': 'Now the base of this right triangle has length A, the X coordinate, and the height of the right triangle is B, the y coordinate.', 'start': 6628.228, 'duration': 10.068}, {'end': 6647.721, 'text': 'If I use the right triangle definition of sine and cosine of theta, this right here is the angle theta.', 'start': 6640.697, 'duration': 7.024}, {'end': 6652.603, 'text': 'then cosine of theta is adjacent over hypotenuse.', 'start': 6647.721, 'duration': 4.882}, {'end': 6658.166, 'text': "So that's a over one, or a.", 'start': 6652.963, 'duration': 5.203}, {'end': 6667.397, 'text': 'Notice that a also represents the x coordinate of this point on the unit circle at angle theta from the x axis.', 'start': 6658.166, 'duration': 9.231}, {'end': 6668.738, 'text': "I'll write that down.", 'start': 6668.017, 'duration': 0.721}, {'end': 6676.863, 'text': "For sine of theta, if I use the right triangle definition, that's opposite over hypotenuse.", 'start': 6670.419, 'duration': 6.444}, {'end': 6682.647, 'text': 'So B over one, which is just B.', 'start': 6677.203, 'duration': 5.444}, {'end': 6689.311, 'text': 'But B also represents the y coordinate of this point in the on the unit circle at angle theta.', 'start': 6682.647, 'duration': 6.664}, {'end': 6696.917, 'text': "for tangent theta, if we use the right side triangle definition, it's opposite over adjacent.", 'start': 6690.953, 'duration': 5.964}, {'end': 6704.161, 'text': "So that's B over a, I can think of that as the y coordinate of the point over the x coordinate of the point.", 'start': 6697.357, 'duration': 6.804}, {'end': 6710.424, 'text': "Now for angles theta that can't be part of a right triangle because they're too big.", 'start': 6705.822, 'duration': 4.602}, {'end': 6712.686, 'text': "they're bigger than 90 degrees, like now.", 'start': 6710.424, 'duration': 2.262}, {'end': 6715.714, 'text': "I'll call this angle here theta.", 'start': 6712.686, 'duration': 3.028}, {'end': 6723.238, 'text': 'I can still use this idea of x and y coordinates to calculate the sine and cosine of theta.', 'start': 6715.714, 'duration': 7.524}, {'end': 6735.744, 'text': 'So if I just mark this point on the end of this line at angle theta, if I mark that to have coordinates x and y, then cosine theta,', 'start': 6724.298, 'duration': 11.446}, {'end': 6742.807, 'text': "I'm still going to define as the x coordinate of this point, sine theta, as the y coordinate.", 'start': 6735.744, 'duration': 7.063}, {'end': 6749.891, 'text': 'and tangent theta as the ratio of the y coordinate over the x coordinate.', 'start': 6743.567, 'duration': 6.324}, {'end': 6761.758, 'text': 'When we use this unit circle definition, we always draw theta starting from the positive x axis and going counterclockwise.', 'start': 6751.992, 'duration': 9.766}, {'end': 6770.864, 'text': "Let's use this unit circle definition to calculate sine, cosine and tangent of this angle phi.", 'start': 6765.08, 'duration': 5.784}], 'summary': 'Unit circle relates right triangles, coordinates, and trigonometric functions for angles up to 90 degrees.', 'duration': 204.62, 'max_score': 6566.244, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/eI4an8aSsgw/pics/eI4an8aSsgw6566244.jpg'}, {'end': 6884.17, 'src': 'embed', 'start': 6814.838, 'weight': 1, 'content': [{'end': 6820.583, 'text': 'This video gives a method for calculating sine, cosine and tangent in terms of the unit circle.', 'start': 6814.838, 'duration': 5.745}, {'end': 6827.849, 'text': 'Starting from the positive x axis, you draw the angle theta going counterclockwise.', 'start': 6822.044, 'duration': 5.805}, {'end': 6837.213, 'text': 'you look at the coordinates of the point on the unit circle where that angle ends.', 'start': 6831.448, 'duration': 5.765}, {'end': 6849.863, 'text': 'And cosine of that angle theta is the x coordinate, sine of theta is the y coordinate, and tangent of theta is the ratio.', 'start': 6839.354, 'duration': 10.509}, {'end': 6858.95, 'text': 'This video gives three properties of the trig function sine and cosine that can be deduced from the unit circle definition.', 'start': 6851.944, 'duration': 7.006}, {'end': 6878.447, 'text': 'Recall that the unit circle definition of sine and cosine for an angle theta is that cosine theta is the x coordinate and sine of theta is the y coordinate for the point on the unit circle at angle theta.', 'start': 6860.379, 'duration': 18.068}, {'end': 6884.17, 'text': 'The first property is what I call the periodic property.', 'start': 6881.468, 'duration': 2.702}], 'summary': 'Video explains sine, cosine, tangent calculation method using unit circle.', 'duration': 69.332, 'max_score': 6814.838, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/eI4an8aSsgw/pics/eI4an8aSsgw6814838.jpg'}, {'end': 7152.778, 'src': 'heatmap', 'start': 6947.659, 'weight': 0.803, 'content': [{'end': 6956.743, 'text': 'then theta plus two pi, the plus two pi adds a full turn around the unit circle to our angle.', 'start': 6947.659, 'duration': 9.084}, {'end': 6958.184, 'text': 'So we end up at the same place.', 'start': 6956.783, 'duration': 1.401}, {'end': 6966.087, 'text': 'Theta and theta plus two pi are just two different names for the same location on the unit circle.', 'start': 6959.544, 'duration': 6.543}, {'end': 6973.29, 'text': 'And since sine and cosine give you the y and x coordinates of that point on the unit circle, they have to have the same value.', 'start': 6966.947, 'duration': 6.343}, {'end': 6985.175, 'text': 'Similarly, if we consider an angle theta, an angle theta minus two pi, the minus two pi means we go the other direction,', 'start': 6974.751, 'duration': 10.424}, {'end': 6989.478, 'text': 'around the unit circle clockwise, we still end up in the same place.', 'start': 6985.175, 'duration': 4.303}, {'end': 6993.661, 'text': 'And therefore cosine of theta minus two pi.', 'start': 6990.059, 'duration': 3.602}, {'end': 6998.485, 'text': 'the x coordinate of that position is the same thing as cosine of theta.', 'start': 6993.661, 'duration': 4.824}, {'end': 7002.488, 'text': 'sine of theta minus two pi is the same thing as sine of theta.', 'start': 6998.485, 'duration': 4.003}, {'end': 7006.871, 'text': 'The same statements hold if we add or subtract multiples of two pi.', 'start': 7003.448, 'duration': 3.423}, {'end': 7013.133, 'text': 'For example, cosine of theta plus four pi is still the same thing as cosine of theta.', 'start': 7007.449, 'duration': 5.684}, {'end': 7019.618, 'text': "This time, we've just gone two turns around the unit circle and still gotten back to the same place.", 'start': 7013.153, 'duration': 6.465}, {'end': 7031.286, 'text': "So if we want to find cosine of five pi, that's the same thing as cosine of pi plus four pi, which is the same thing as cosine of pi.", 'start': 7021.279, 'duration': 10.007}, {'end': 7036.99, 'text': 'Thinking about the unit circle, pi is halfway around the unit circle.', 'start': 7032.987, 'duration': 4.003}, {'end': 7046.238, 'text': 'So cosine of pi means the x coordinate of this point right here, well, that point has coordinates negative 10.', 'start': 7037.694, 'duration': 8.544}, {'end': 7049.099, 'text': 'So cosine of pi must be negative one.', 'start': 7046.238, 'duration': 2.861}, {'end': 7054.201, 'text': 'If I want to take sine of negative 420 degrees or that sine of negative 360 degrees, minus 60 degrees,', 'start': 7050.24, 'duration': 3.961}, {'end': 7057.022, 'text': 'which is the same thing as sine of minus 60 degrees.', 'start': 7054.201, 'duration': 2.821}, {'end': 7077.934, 'text': 'Thinking about the unit circle minus 60 degrees means I start at the positive x axis and go clockwise by 60 degrees that lands me about right here.', 'start': 7066.305, 'duration': 11.629}, {'end': 7094.408, 'text': "And so that's one of the special angles that has an x coordinate of one half a y coordinate of negative root three over two and therefore sine of negative 60 is negative root three over to the y coordinate.", 'start': 7078.595, 'duration': 15.813}, {'end': 7097.5, 'text': 'The next property I call the even odd property.', 'start': 7095.339, 'duration': 2.161}, {'end': 7106.743, 'text': 'It says that cosine is an even function, which means that cosine of negative theta is the same thing as cosine of theta.', 'start': 7098.62, 'duration': 8.123}, {'end': 7116.147, 'text': 'While sine is an odd function, which means that sine of negative theta is the negative of sine of theta.', 'start': 7107.884, 'duration': 8.263}, {'end': 7118.988, 'text': 'To see why this is true.', 'start': 7117.227, 'duration': 1.761}, {'end': 7123.789, 'text': "let's look at an angle theta and the angle negative theta.", 'start': 7118.988, 'duration': 4.801}, {'end': 7130.748, 'text': 'a negative angle means you go in the clockwise instead of counterclockwise direction from the positive x axis.', 'start': 7123.789, 'duration': 6.959}, {'end': 7138.612, 'text': 'The coordinates of this point, by definition, are cosine theta sine theta,', 'start': 7133.069, 'duration': 5.543}, {'end': 7145.395, 'text': 'whereas the coordinates of this point are cosine negative theta sine of negative theta.', 'start': 7138.612, 'duration': 6.783}, {'end': 7152.778, 'text': 'But by symmetry, these two points have the exact same x coordinate.', 'start': 7148.436, 'duration': 4.342}], 'summary': 'Adding or subtracting 2pi to theta yields same cosine and sine values, and cosine is even while sine is odd.', 'duration': 205.119, 'max_score': 6947.659, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/eI4an8aSsgw/pics/eI4an8aSsgw6947659.jpg'}, {'end': 7123.789, 'src': 'embed', 'start': 7095.339, 'weight': 2, 'content': [{'end': 7097.5, 'text': 'The next property I call the even odd property.', 'start': 7095.339, 'duration': 2.161}, {'end': 7106.743, 'text': 'It says that cosine is an even function, which means that cosine of negative theta is the same thing as cosine of theta.', 'start': 7098.62, 'duration': 8.123}, {'end': 7116.147, 'text': 'While sine is an odd function, which means that sine of negative theta is the negative of sine of theta.', 'start': 7107.884, 'duration': 8.263}, {'end': 7118.988, 'text': 'To see why this is true.', 'start': 7117.227, 'duration': 1.761}, {'end': 7123.789, 'text': "let's look at an angle theta and the angle negative theta.", 'start': 7118.988, 'duration': 4.801}], 'summary': 'Cosine is an even function, sine is an odd function.', 'duration': 28.45, 'max_score': 7095.339, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/eI4an8aSsgw/pics/eI4an8aSsgw7095339.jpg'}, {'end': 7248.934, 'src': 'embed', 'start': 7220.199, 'weight': 3, 'content': [{'end': 7229.464, 'text': 'The last property on this video is the Pythagorean property, which says that cosine of theta squared plus sine of theta squared is equal to one.', 'start': 7220.199, 'duration': 9.265}, {'end': 7240.711, 'text': 'A lot of times this property is written with the shorthand notation cosine squared theta plus sine squared theta equals one.', 'start': 7229.784, 'duration': 10.927}, {'end': 7248.934, 'text': 'But this notation cosine squared theta just means you take cosine of theta and square it.', 'start': 7241.428, 'duration': 7.506}], 'summary': 'Pythagorean property: cos^2(theta) + sin^2(theta) = 1.', 'duration': 28.735, 'max_score': 7220.199, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/eI4an8aSsgw/pics/eI4an8aSsgw7220199.jpg'}, {'end': 7337.173, 'src': 'embed', 'start': 7303.697, 'weight': 4, 'content': [{'end': 7309.36, 'text': 'Pythagorean property is handy for computing values of cosine, given values of sine and vice versa.', 'start': 7303.697, 'duration': 5.663}, {'end': 7313.662, 'text': "And this problem, we're told that sine of t is negative two sevenths.", 'start': 7310.46, 'duration': 3.202}, {'end': 7316.663, 'text': 'And t is an angle that lies in quadrant three.', 'start': 7314.142, 'duration': 2.521}, {'end': 7325.166, 'text': 'When we say the angle lies in quadrant three, that means the terminal side of the angle lies here in quadrant three.', 'start': 7318.143, 'duration': 7.023}, {'end': 7337.173, 'text': 'One way to find cosine of t is to use the fact that cosine squared t plus sine squared t is equal to one.', 'start': 7327.387, 'duration': 9.786}], 'summary': 'Use pythagorean property to find cosine of t given sine of t as -2/7 in quadrant three.', 'duration': 33.476, 'max_score': 7303.697, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/eI4an8aSsgw/pics/eI4an8aSsgw7303697.jpg'}], 'start': 6499.24, 'title': 'Trigonometry fundamentals', 'summary': 'Covers special angles 30, 45, and 60 degrees, defining sine and cosine in terms of the unit circle, emphasizes their use to avoid computation errors, explains the relationship between right triangles and the unit circle, gives methods for calculating sine, cosine, and tangent, and discusses the pythagorean property and its application in solving trigonometric problems.', 'chapters': [{'end': 6563.341, 'start': 6499.24, 'title': 'Special angles and unit circle', 'summary': 'Introduces the sine and cosine of special angles 30, 45, and 60 degrees, defining them in terms of points on the unit circle and emphasizing their use to avoid computation errors for angles beyond the first quadrant.', 'duration': 64.101, 'highlights': ['The video computes the sine and cosine of special angles 30, 45, and 60 degrees, emphasizing their significance in trigonometry.', 'It defines sine and cosine in terms of points on the unit circle, providing a visual understanding of these trigonometric functions.', 'The chapter highlights the limitations of using right triangles to compute sine and cosine, particularly for angles beyond the first quadrant.']}, {'end': 7217.317, 'start': 6566.244, 'title': 'Unit circle and trig functions', 'summary': 'Explains the relationship between right triangles and the unit circle, and gives methods for calculating sine, cosine and tangent in terms of the unit circle, as well as properties of sine and cosine functions, including periodic and even-odd properties.', 'duration': 651.073, 'highlights': ['The video discusses the relationship between right triangles and the unit circle, providing a method for calculating sine, cosine and tangent in terms of the unit circle.', 'The chapter covers properties of sine and cosine functions, including the periodic property stating that the values of cosine and sine are periodic with period two pi, and the even-odd property stating that cosine is even while sine is odd.', 'It details the method for calculating sine, cosine and tangent in terms of the unit circle, and provides examples with detailed calculations and explanations.']}, {'end': 7477.144, 'start': 7220.199, 'title': 'Pythagorean property and trigonometric problem', 'summary': 'Discusses the pythagorean property, its application in finding cosine of an angle in quadrant three, and the alternative method of solving the trigonometric problem using the pythagorean theorem, ultimately obtaining the same result.', 'duration': 256.945, 'highlights': ['The Pythagorean property, cosine squared theta plus sine squared theta equals one, is derived from the Pythagorean theorem, providing a useful tool for computing values of cosine and sine, as demonstrated in a problem where cosine of t is found to be negative square root of 45 over seven.', 'The application of the Pythagorean property in finding cosine of an angle in quadrant three is illustrated, with the solution involving the computation of the square root of 45 over seven, ultimately yielding a negative value in accordance with the quadrant.', 'An alternative method of solving the trigonometric problem using the Pythagorean theorem directly is presented, yielding the same result for the cosine of the given angle t in quadrant three.']}], 'duration': 977.904, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/eI4an8aSsgw/pics/eI4an8aSsgw6499240.jpg', 'highlights': ['The video computes the sine and cosine of special angles 30, 45, and 60 degrees, emphasizing their significance in trigonometry.', 'It defines sine and cosine in terms of points on the unit circle, providing a visual understanding of these trigonometric functions.', 'The chapter covers properties of sine and cosine functions, including the periodic property stating that the values of cosine and sine are periodic with period two pi, and the even-odd property stating that cosine is even while sine is odd.', 'The Pythagorean property, cosine squared theta plus sine squared theta equals one, is derived from the Pythagorean theorem, providing a useful tool for computing values of cosine and sine, as demonstrated in a problem where cosine of t is found to be negative square root of 45 over seven.', 'The application of the Pythagorean property in finding cosine of an angle in quadrant three is illustrated, with the solution involving the computation of the square root of 45 over seven, ultimately yielding a negative value in accordance with the quadrant.', 'The video discusses the relationship between right triangles and the unit circle, providing a method for calculating sine, cosine and tangent in terms of the unit circle.', 'It details the method for calculating sine, cosine and tangent in terms of the unit circle, and provides examples with detailed calculations and explanations.', 'The chapter highlights the limitations of using right triangles to compute sine and cosine, particularly for angles beyond the first quadrant.', 'An alternative method of solving the trigonometric problem using the Pythagorean theorem directly is presented, yielding the same result for the cosine of the given angle t in quadrant three.']}, {'end': 8879.056, 'segs': [{'end': 7742.56, 'src': 'embed', 'start': 7707.853, 'weight': 0, 'content': [{'end': 7709.994, 'text': "Next, let's look at domain and range.", 'start': 7707.853, 'duration': 2.141}, {'end': 7714.316, 'text': 'The domain of sine and cosine is all real numbers.', 'start': 7710.895, 'duration': 3.421}, {'end': 7716.517, 'text': 'Alright, that is negative infinity to infinity.', 'start': 7714.556, 'duration': 1.961}, {'end': 7719.819, 'text': 'But the range is just from negative one to one.', 'start': 7716.938, 'duration': 2.881}, {'end': 7724.622, 'text': 'That makes sense, because sine and cosine come from the unit circle.', 'start': 7721.2, 'duration': 3.422}, {'end': 7728.664, 'text': 'The input values for the domain come from angles.', 'start': 7725.542, 'duration': 3.122}, {'end': 7735.027, 'text': 'And you can use any numbers and angle positive negative as big as you want, just by wrapping a lot of times around the circle.', 'start': 7729.244, 'duration': 5.783}, {'end': 7742.56, 'text': 'The output values for the range, that is the actual values of sine and cosine, come from the coordinates on the unit circle.', 'start': 7735.816, 'duration': 6.744}], 'summary': 'The domain of sine and cosine is all real numbers from negative infinity to infinity, but the range is just from negative one to one, as they come from the unit circle.', 'duration': 34.707, 'max_score': 7707.853, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/eI4an8aSsgw/pics/eI4an8aSsgw7707853.jpg'}, {'end': 7931.592, 'src': 'heatmap', 'start': 7729.244, 'weight': 0.878, 'content': [{'end': 7735.027, 'text': 'And you can use any numbers and angle positive negative as big as you want, just by wrapping a lot of times around the circle.', 'start': 7729.244, 'duration': 5.783}, {'end': 7742.56, 'text': 'The output values for the range, that is the actual values of sine and cosine, come from the coordinates on the unit circle.', 'start': 7735.816, 'duration': 6.744}, {'end': 7747.143, 'text': "And those coordinates can't be any bigger than 1 or any smaller than negative 1.", 'start': 7742.96, 'duration': 4.183}, {'end': 7748.183, 'text': 'So that gives us our range.', 'start': 7747.143, 'duration': 1.04}, {'end': 7757.989, 'text': "As far as even and odd behavior, you can tell from the graph, here's cosine, that it's symmetric with respect to the y-axis, and so it must be even.", 'start': 7749.524, 'duration': 8.465}, {'end': 7762.892, 'text': 'Whereas the graph of sine is symmetric with respect to the origin and must be odd.', 'start': 7758.509, 'duration': 4.383}, {'end': 7770.784, 'text': 'The absolute maximum values of these two functions is one and the absolute minimum value is negative one.', 'start': 7764.022, 'duration': 6.762}, {'end': 7778.506, 'text': 'We can also use the words midline, amplitude and period to describe these two functions.', 'start': 7773.485, 'duration': 5.021}, {'end': 7786.028, 'text': 'The midline is the horizontal line halfway in between the maximum and minimum points.', 'start': 7779.586, 'duration': 6.442}, {'end': 7789.529, 'text': 'Here, the midline is y equals zero.', 'start': 7786.928, 'duration': 2.601}, {'end': 7796.286, 'text': 'The amplitude is the vertical distance between a maximum point and the midline.', 'start': 7791.004, 'duration': 5.282}, {'end': 7807.632, 'text': 'You can also think of the amplitude as the vertical distance between a minimum point and the midline or as half the vertical distance between a min point and a max point.', 'start': 7797.547, 'duration': 10.085}, {'end': 7812.514, 'text': 'For the cosine function and the sine function, the amplitude is one.', 'start': 7808.912, 'duration': 3.602}, {'end': 7817.812, 'text': 'A periodic function is a function that repeats at regular horizontal intervals.', 'start': 7813.809, 'duration': 4.003}, {'end': 7823.557, 'text': 'The horizontal length of the smallest repeating unit is called the period.', 'start': 7818.633, 'duration': 4.924}, {'end': 7828.601, 'text': 'For y equals cosine of x, the period is two pi.', 'start': 7824.658, 'duration': 3.943}, {'end': 7839.57, 'text': 'Notice that the period is the horizontal distance between successive peaks, or maximum points, or between successive troughs, or minimum points.', 'start': 7830.242, 'duration': 9.328}, {'end': 7853.47, 'text': 'Algebraically, we can write cosine of x plus two pi equals cosine of x and sine of x plus two pi equals sine of x,', 'start': 7841.266, 'duration': 12.204}, {'end': 7859.852, 'text': 'to indicate that the functions repeat themselves over an interval of two pi and have a period of two pi.', 'start': 7853.47, 'duration': 6.382}, {'end': 7867.814, 'text': 'In this video, we graphed y equals cosine of x and y equals sine of x.', 'start': 7861.312, 'duration': 6.502}, {'end': 7880.452, 'text': 'and observe that they both have a midline at y equals zero, an amplitude of one and a period of two pi.', 'start': 7870.069, 'duration': 10.383}, {'end': 7891.296, 'text': 'sine u sort all functions are functions that are related to sine and cosine by transformations like stretching and shrinking and shifting.', 'start': 7883.373, 'duration': 7.923}, {'end': 7894.657, 'text': 'This video is about graphing these functions.', 'start': 7892.536, 'duration': 2.121}, {'end': 7901.225, 'text': "Let's start by graphing the function y equals three sine of two x.", 'start': 7895.782, 'duration': 5.443}, {'end': 7905.308, 'text': 'This function is related to the function y equals sine x.', 'start': 7901.225, 'duration': 4.083}, {'end': 7906.388, 'text': "So I'll graph that first.", 'start': 7905.308, 'duration': 1.08}, {'end': 7915.074, 'text': 'Now the three on the outside stretches this graph vertically by a factor of three,', 'start': 7907.689, 'duration': 7.385}, {'end': 7920.457, 'text': 'while the two on the inside compresses it horizontally by a factor of one half.', 'start': 7915.074, 'duration': 5.383}, {'end': 7931.592, 'text': 'If instead, I want to graph y equals three sine two x plus one, this plus one on the outside shifts everything up by one unit.', 'start': 7921.844, 'duration': 9.748}], 'summary': 'Graphed sine and cosine functions with properties like range, even/odd behavior, amplitude, and period; then related functions with transformations.', 'duration': 202.348, 'max_score': 7729.244, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/eI4an8aSsgw/pics/eI4an8aSsgw7729244.jpg'}, {'end': 7778.506, 'src': 'embed', 'start': 7749.524, 'weight': 1, 'content': [{'end': 7757.989, 'text': "As far as even and odd behavior, you can tell from the graph, here's cosine, that it's symmetric with respect to the y-axis, and so it must be even.", 'start': 7749.524, 'duration': 8.465}, {'end': 7762.892, 'text': 'Whereas the graph of sine is symmetric with respect to the origin and must be odd.', 'start': 7758.509, 'duration': 4.383}, {'end': 7770.784, 'text': 'The absolute maximum values of these two functions is one and the absolute minimum value is negative one.', 'start': 7764.022, 'duration': 6.762}, {'end': 7778.506, 'text': 'We can also use the words midline, amplitude and period to describe these two functions.', 'start': 7773.485, 'duration': 5.021}], 'summary': 'Cosine is even, sine is odd. absolute max/min values are 1 and -1.', 'duration': 28.982, 'max_score': 7749.524, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/eI4an8aSsgw/pics/eI4an8aSsgw7749524.jpg'}, {'end': 8218.76, 'src': 'embed', 'start': 8190.063, 'weight': 2, 'content': [{'end': 8197.206, 'text': "Let's take a moment to look at midline, amplitude, and period.", 'start': 8190.063, 'duration': 7.143}, {'end': 8211.394, 'text': 'For the original parent function y equals sine of x and our final transformed function y equals three sine of two times quantity x minus pi over four minus one.', 'start': 8198.202, 'duration': 13.192}, {'end': 8218.76, 'text': 'Our original sine function has midline at y equals zero amplitude of one and period of two pi.', 'start': 8212.575, 'duration': 6.185}], 'summary': 'Analyzing sine function with midline, amplitude of 1, and period of 2π.', 'duration': 28.697, 'max_score': 8190.063, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/eI4an8aSsgw/pics/eI4an8aSsgw8190063.jpg'}, {'end': 8501.944, 'src': 'embed', 'start': 8476.658, 'weight': 3, 'content': [{'end': 8483.443, 'text': 'Better rewrite this horizontal shift of six units to the left.', 'start': 8476.658, 'duration': 6.785}, {'end': 8489.197, 'text': 'The horizontal shift is sometimes called the phase shift.', 'start': 8485.575, 'duration': 3.622}, {'end': 8494.66, 'text': "And that's all for graphs of sinusoidal functions.", 'start': 8492.339, 'duration': 2.321}, {'end': 8501.944, 'text': 'This video is about graphing the trig functions tangent, secant, cotangent, and cosecant.', 'start': 8496.181, 'duration': 5.763}], 'summary': 'Graph trig functions: shift -6 units left, cover tangent, secant, cotangent, cosecant.', 'duration': 25.286, 'max_score': 8476.658, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/eI4an8aSsgw/pics/eI4an8aSsgw8476658.jpg'}], 'start': 7478.786, 'title': 'Trig functions & graphs', 'summary': 'Discusses trigonometric functions, their properties, and graphing processes of y equals cosine t and y equals sine t in radians, covering periodic, even odd, and pythagorean properties, and transformations such as stretching, shrinking, and shifting.', 'chapters': [{'end': 7748.183, 'start': 7478.786, 'title': 'Trig functions & graphs', 'summary': 'Discusses the properties of trigonometric functions, including periodic, even odd, and pythagorean properties, and explains the graphing process of y equals cosine t and y equals sine t in radians, highlighting the domain and range of sine and cosine functions.', 'duration': 269.397, 'highlights': ['The domain of sine and cosine is all real numbers, while the range is from -1 to 1, stemming from the unit circle coordinates.', 'The graph of cosine is similar to the graph of sine, with cosine being a shifted version of sine by pi/2, and sine being constructed from the cosine graph by shifting it to the right by pi/2.', 'The video demonstrates the plotting of points for cosine and sine functions and explains the repetition of cosine values as t values change, and extends the graph for sine through repetition.']}, {'end': 8045.101, 'start': 7749.524, 'title': 'Graphing sine and cosine functions', 'summary': 'Explains the properties of sine and cosine functions, including their even and odd behavior, midline, amplitude, and period, and demonstrates how these properties change with transformations, such as stretching, shrinking, and shifting.', 'duration': 295.577, 'highlights': ['The functions cosine and sine are symmetric with respect to the y-axis and the origin, and have absolute maximum values of one and absolute minimum value of negative one.', 'The midline, amplitude, and period are key properties used to describe the functions, with the midline being y equals zero, and the amplitude and period both equal to one for the cosine and sine functions.', 'Transformations such as stretching and shrinking affect the amplitude and period of the functions, while shifting impacts the midline, as demonstrated in the examples of y equals three sine of two x and y equals three sine two x plus one.']}, {'end': 8324.536, 'start': 8046.301, 'title': 'Graphing transformed sine function', 'summary': 'Explains the process of graphing a transformed sine function, including vertical and horizontal stretching, shifting, and period adjustments, with specific examples and formulas provided for each transformation.', 'duration': 278.235, 'highlights': ['The function y equals three times sine of two times quantity x minus pi over four is closely related to y equals 3 sine of 2x.', 'The original sine function has midline at y equals zero, amplitude of one, and period of two pi, while the transformed function y equals three sine of two times quantity x minus pi over four minus one has an amplitude of three, midline at y equals negative one, and period of pi.', 'The horizontal shift, or phase shift, of the transform function y equals three sine two x minus pi over four minus one can also be represented as y equals three sine two x minus pi over two minus one.']}, {'end': 8879.056, 'start': 8324.536, 'title': 'Graphing trig functions', 'summary': 'Explains how to graph sinusoidal functions, particularly focusing on the amplitude, period, and horizontal shift of cosine and sine functions, and then delves into the graphing of tangent and secant functions, detailing their period, x-intercepts, vertical asymptotes, domain, and range.', 'duration': 554.52, 'highlights': ['The chapter explains how to graph sinusoidal functions, particularly focusing on the amplitude, period, and horizontal shift of cosine and sine functions.', 'The chapter then delves into the graphing of tangent and secant functions, detailing their period, x-intercepts, vertical asymptotes, domain, and range.']}], 'duration': 1400.27, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/eI4an8aSsgw/pics/eI4an8aSsgw7478786.jpg', 'highlights': ['The domain of sine and cosine is all real numbers, while the range is from -1 to 1, stemming from the unit circle coordinates.', 'The functions cosine and sine are symmetric with respect to the y-axis and the origin, and have absolute maximum values of one and absolute minimum value of negative one.', 'The original sine function has midline at y equals zero, amplitude of one, and period of two pi, while the transformed function y equals three sine of two times quantity x minus pi over four minus one has an amplitude of three, midline at y equals negative one, and period of pi.', 'The chapter explains how to graph sinusoidal functions, particularly focusing on the amplitude, period, and horizontal shift of cosine and sine functions.']}, {'end': 11587.898, 'segs': [{'end': 9035.258, 'src': 'embed', 'start': 9001.313, 'weight': 0, 'content': [{'end': 9009.424, 'text': "In fact, if I draw the graph of sine x in between, You can see how it kind of bounces off because it's the reciprocal.", 'start': 9001.313, 'duration': 8.111}, {'end': 9014.87, 'text': 'I encourage you to memorize the general shape of these graphs.', 'start': 9011.946, 'duration': 2.924}, {'end': 9026.663, 'text': "You can always figure out the details by thinking about how they're related to the graphs of cosine of x and sine x.", 'start': 9015.19, 'duration': 11.473}, {'end': 9035.258, 'text': 'This video is about the graphs of transformations of tangent secant cotangent and cosecant.', 'start': 9029.656, 'duration': 5.602}], 'summary': 'Learn the general shape of trigonometric graphs and their transformations.', 'duration': 33.945, 'max_score': 9001.313, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/eI4an8aSsgw/pics/eI4an8aSsgw9001313.jpg'}, {'end': 9594.99, 'src': 'embed', 'start': 9566.96, 'weight': 2, 'content': [{'end': 9575.663, 'text': 'The graph of the inverse of a function can be found by flipping the graph of the original function over the line y equals x.', 'start': 9566.96, 'duration': 8.703}, {'end': 9578.364, 'text': "I've drawn the flipped graph with this blue dotted line.", 'start': 9575.663, 'duration': 2.701}, {'end': 9585.466, 'text': "But you'll notice that the blue dotted line is not the graph of a function because it violates the vertical line test.", 'start': 9579.964, 'duration': 5.502}, {'end': 9594.99, 'text': "So in order to get a function that's the inverse of y equals sine x, we need to restrict the domain of sine of x.", 'start': 9587.168, 'duration': 7.822}], 'summary': 'To find the inverse of a function, flip the graph over y=x. the inverse of y=sin(x) requires restricting its domain.', 'duration': 28.03, 'max_score': 9566.96, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/eI4an8aSsgw/pics/eI4an8aSsgw9566960.jpg'}, {'end': 10637.044, 'src': 'embed', 'start': 10612.599, 'weight': 1, 'content': [{'end': 10618.583, 'text': 'To get a decimal approximation for one of these angles, I can take our cosine of both sides of my equation.', 'start': 10612.599, 'duration': 5.984}, {'end': 10627.898, 'text': 'This gives me t equals our cosine of one third which, using my calculator on radian mode,', 'start': 10620.353, 'duration': 7.545}, {'end': 10634.102, 'text': 'I get a decimal approximation of 1.2310 up to four decimal places.', 'start': 10627.898, 'duration': 6.204}, {'end': 10637.044, 'text': "It's important to use radian mode here.", 'start': 10634.963, 'duration': 2.081}], 'summary': 'Using cosine, t equals the cosine of one third, giving a decimal approximation of 1.2310 in radian mode.', 'duration': 24.445, 'max_score': 10612.599, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/eI4an8aSsgw/pics/eI4an8aSsgw10612599.jpg'}, {'end': 11253.249, 'src': 'embed', 'start': 11222.398, 'weight': 3, 'content': [{'end': 11225.299, 'text': 'We can say that the solution set is all real numbers.', 'start': 11222.398, 'duration': 2.901}, {'end': 11236.874, 'text': 'This second equation is called an identity because it holds for all values of the variable.', 'start': 11228.08, 'duration': 8.794}, {'end': 11245.202, 'text': 'The first equation, on the other hand, is not an identity because it only holds for some values of x and not all values.', 'start': 11238.456, 'duration': 6.746}, {'end': 11253.249, 'text': 'Please pause the video for a moment and try to decide which of the following three equations are identities.', 'start': 11246.963, 'duration': 6.286}], 'summary': 'The solution set is all real numbers. the second equation is an identity, holding for all values of the variable, while the first equation is not an identity, holding only for some values of x.', 'duration': 30.851, 'max_score': 11222.398, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/eI4an8aSsgw/pics/eI4an8aSsgw11222398.jpg'}], 'start': 8879.997, 'title': 'Trigonometric functions and equations', 'summary': 'Discusses graphs of secant, cosecant, tangent, and cotangent functions, their periods, ranges, domains, and transformation rules with examples. it also covers inverse trig functions, their graphs, domains, ranges, and relationships to original functions, as well as solving trig equations with a calculator and quadratic functions and identity equations.', 'chapters': [{'end': 9540.879, 'start': 8879.997, 'title': 'Graphs of trigonometric functions', 'summary': 'Discusses the graphs of secant, cosecant, tangent, and cotangent functions, highlighting their periods, ranges, domains, vertical asymptotes, x-intercepts, and transformation rules with examples, providing a comprehensive understanding of their graphical representations and transformations.', 'duration': 660.882, 'highlights': ['The chapter discusses the graphs of secant, cosecant, tangent, and cotangent functions, highlighting their periods, ranges, domains, vertical asymptotes, x-intercepts, and transformation rules with examples.', 'The vertical asymptotes occur at values of the form pi over two k, where k is an odd integer, and the domain excludes these values.', 'The period of the graph of secant is 2 pi, and it has a range from negative infinity to negative 1 inclusive, and from 1 to infinity.', 'The graphs of cosecant and secant are related to the graphs of sine and cosine, respectively, through the concept of reciprocals.', 'The transformation rules for graphing functions such as scaling, shifting, and phase shifting are demonstrated with the graph of cosecant, providing a comprehensive understanding of these concepts.']}, {'end': 9942.661, 'start': 9540.879, 'title': 'Inverse trig functions & graphs', 'summary': 'Defines the standard inverse trig functions, sine inverse, cosine inverse, and tan inverse, explaining their graphs, domains, ranges, and their relationship to the original functions.', 'duration': 401.782, 'highlights': ['The inverse of a function is found by flipping the graph of the original function over the line y equals x, and by restricting the domain of the original function to obtain a proper inverse function.', 'The domain of the inverse sine function is from -1 to 1 and the range is from -pi/2 to pi/2, and it reverses the roles of y and x.', 'The inverse cosine function has a domain from -1 to 1 and range from 0 to pi, and it reverses the roles of y and x similar to the inverse sine function.', 'The inverse tangent function involves taking just one piece of the tangent function to obtain a proper inverse function, and it has vertical asymptotes as part of its graph.']}, {'end': 10555.691, 'start': 9943.939, 'title': 'Inverse trig functions & solving trig equations', 'summary': 'Covers the conventions for inverting the tangent function, the domain and range of restricted tangent function, and solving trig equations involving cosine and tangent, leading to a general formula for all solutions.', 'duration': 611.752, 'highlights': ['The restricted tan function has domain from negative pi over two to pi over two, and the range is from negative infinity to infinity.', 'Arctan of x has domain from negative infinity to infinity and range from negative pi over 2 to pi over 2.', 'Finding all solutions for the equation 2 cosine x plus 1 equals 0 in the interval from 0 to 2 pi, and then getting a general formula for all solutions.', 'Solving a trig equation involving tangent and using the unit circle to find principal solutions, then expressing all solutions in a more efficient form.', 'Solving basic trig equations by isolating sine, cosine, or tangent, finding principal solutions using the unit circle, and obtaining all solutions by adding multiples of two pi or pi.']}, {'end': 11150.193, 'start': 10557.051, 'title': 'Solving trig equations with calculator', 'summary': 'Covers solving trig equations using a calculator, finding decimal approximations for solutions, identifying multiple solutions within a given interval, and graphically representing solutions for cosine, sine, and tangent equations.', 'duration': 593.142, 'highlights': ['The chapter emphasizes finding decimal approximations for solutions using a calculator, such as obtaining the decimal approximation of 1.2310 for the equation 2 cosine t equals 1 minus cosine t.', 'It explains the concept of multiple solutions within a given interval, as seen in the case of cosine t equals 1 third, where there are two different solutions within the interval 0 to 2 pi: 1.2310 and 5.0522 radians.', 'The chapter demonstrates the use of symmetry to find the second angle for a solution, showcasing the approach of using pi minus the first angle for sine equations.', 'It concludes by providing general methods for finding all solutions to cosine, sine, and tangent equations, highlighting the applicability of the discussed methods to more complex examples.']}, {'end': 11587.898, 'start': 11151.354, 'title': 'Quadratic functions and identity equations', 'summary': 'Covers solving quadratic equations, distinguishing between identities and non-identities, and using algebraic and graphical methods for verification.', 'duration': 436.544, 'highlights': ['The chapter covers solving quadratic equations, distinguishing between identities and non-identities, and using algebraic and graphical methods for verification.', 'The second equation is an identity, as it holds for all values of the variable, backed by evidence from plugging in numbers and graphical analysis.', 'The first equation is not an identity, as it does not hold for all values of the variable, demonstrated through examples of specific x values that do not satisfy the equation.', 'An algebraic verification using the Pythagorean identity and rewriting of terms proved equation C to be an identity for all values of x.']}], 'duration': 2707.901, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/eI4an8aSsgw/pics/eI4an8aSsgw8879997.jpg', 'highlights': ['The chapter discusses the graphs of secant, cosecant, tangent, and cotangent functions, highlighting their periods, ranges, domains, vertical asymptotes, x-intercepts, and transformation rules with examples.', 'The chapter emphasizes finding decimal approximations for solutions using a calculator, such as obtaining the decimal approximation of 1.2310 for the equation 2 cosine t equals 1 minus cosine t.', 'The inverse of a function is found by flipping the graph of the original function over the line y equals x, and by restricting the domain of the original function to obtain a proper inverse function.', 'The chapter covers solving quadratic equations, distinguishing between identities and non-identities, and using algebraic and graphical methods for verification.', 'The transformation rules for graphing functions such as scaling, shifting, and phase shifting are demonstrated with the graph of cosecant, providing a comprehensive understanding of these concepts.']}, {'end': 12467.27, 'segs': [{'end': 11643.518, 'src': 'embed', 'start': 11617.04, 'weight': 1, 'content': [{'end': 11623.764, 'text': 'The second one says tan squared theta plus one equals secant squared theta.', 'start': 11617.04, 'duration': 6.724}, {'end': 11631.248, 'text': 'And the third one goes cotangent squared theta plus one equals cosecant squared theta.', 'start': 11623.784, 'duration': 7.464}, {'end': 11637.573, 'text': "Let's start by proving that cosine squared theta plus sine squared theta equals one.", 'start': 11632.83, 'duration': 4.743}, {'end': 11643.518, 'text': "I'll do this by drawing the unit circle with a right triangle inside it.", 'start': 11639.135, 'duration': 4.383}], 'summary': 'Proving trigonometric identities, such as tan^2+1=sec^2, cot^2+1=csc^2, and cos^2+sin^2=1.', 'duration': 26.478, 'max_score': 11617.04, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/eI4an8aSsgw/pics/eI4an8aSsgw11617040.jpg'}, {'end': 11733.631, 'src': 'embed', 'start': 11705.876, 'weight': 3, 'content': [{'end': 11711.279, 'text': 'I can rewrite that as cosine squared theta plus sine squared theta equals one.', 'start': 11705.876, 'duration': 5.403}, {'end': 11718.325, 'text': 'since one squared is one and cosine squared theta is just a shorthand notation for cosine theta squared.', 'start': 11711.883, 'duration': 6.442}, {'end': 11726.628, 'text': 'That completes the proof of the first Pythagorean identity, at least in the case when the angle theta is in the first quadrant.', 'start': 11720.006, 'duration': 6.622}, {'end': 11733.631, 'text': 'In the case when the angle lies in a different quadrant, you can use symmetry to argue the same identity holds.', 'start': 11728.109, 'duration': 5.522}], 'summary': 'Pythagorean identity: cosine^2 + sine^2 = 1, proven for first quadrant', 'duration': 27.755, 'max_score': 11705.876, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/eI4an8aSsgw/pics/eI4an8aSsgw11705876.jpg'}, {'end': 12069.171, 'src': 'embed', 'start': 12034.98, 'weight': 0, 'content': [{'end': 12040.403, 'text': 'I encourage you to memorize the two formulas for the sine of a sum of angles and the cosine of a sum of angles.', 'start': 12034.98, 'duration': 5.423}, {'end': 12046.346, 'text': "Once you do, it's easy to figure out the sine and cosine of a difference of two angles.", 'start': 12041.043, 'duration': 5.303}, {'end': 12058.153, 'text': 'One way to do this is to think of sine of a minus b as sine of a plus negative b, and then use the angle sum formula.', 'start': 12047.487, 'duration': 10.666}, {'end': 12069.171, 'text': 'So this works out to sine cosine plus cosine sine.', 'start': 12059.113, 'duration': 10.058}], 'summary': 'Memorize sine and cosine sum formulas for easy angle calculations.', 'duration': 34.191, 'max_score': 12034.98, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/eI4an8aSsgw/pics/eI4an8aSsgw12034980.jpg'}], 'start': 11587.898, 'title': 'Trig identities & angle formulas', 'summary': 'Covers proving pythagorean trigonometric identities, sum and difference formulas, and the use of angle sum formulas to find exact values and decimal approximations.', 'chapters': [{'end': 12140.44, 'start': 11587.898, 'title': 'Proving trig identities & sum/difference formulas', 'summary': 'Covers the proof of pythagorean trigonometric identities and the sum and difference formulas, including examples and a mnemonic for remembering the formulas.', 'duration': 552.542, 'highlights': ['The chapter begins by proving the Pythagorean identity cosine squared theta plus sine squared theta equals one, using the unit circle and the Pythagorean theorem, demonstrating the fundamental concept of trigonometry.', 'It then proceeds to prove the tan squared theta plus one equals secant squared theta identity by manipulating the first Pythagorean identity and using algebraic manipulations, showcasing the interrelation between different trigonometric identities.', 'The chapter also proves the third Pythagorean identity cotangent squared theta plus one equals cosecant squared theta using similar algebraic manipulations and trigonometric relationships, further establishing the connection between different trigonometric identities.', 'The chapter concludes with the explanation and mnemonic for remembering the sum and difference formulas for computing sine and cosine of the sum and difference of two angles, providing a practical way to memorize and apply these important formulas in trigonometry.']}, {'end': 12467.27, 'start': 12141.601, 'title': 'Angle sum formulas in trigonometry', 'summary': 'Demonstrates the use of angle sum formulas to find the exact value for the sine of 105 degrees, and to compute the cosine of v plus w using given values of cosine v and w, resulting in a decimal approximation of 0.3187.', 'duration': 325.669, 'highlights': ['The sine of 105 degrees is the sine of 60 plus 45, resulting in root 6 plus root 2 over 4.', 'Computing the cosine of v plus w with given values of cosine v and w results in a decimal approximation of 0.3187.', 'The chapter gives a geometric proof of the angle-sum formulas for computing sine of A plus B and cosine of A plus B.']}], 'duration': 879.372, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/eI4an8aSsgw/pics/eI4an8aSsgw11587898.jpg', 'highlights': ['The chapter concludes with mnemonic for sum and difference formulas for sine and cosine.', 'The chapter proves cotangent squared theta plus one equals cosecant squared theta.', 'The chapter proves tan squared theta plus one equals secant squared theta.', 'The chapter proves cosine squared theta plus sine squared theta equals one.']}, {'end': 13844.716, 'segs': [{'end': 12691.211, 'src': 'embed', 'start': 12663.786, 'weight': 2, 'content': [{'end': 12667.748, 'text': "So I think that's a pretty great geometric proof of the angle sum formulas.", 'start': 12663.786, 'duration': 3.962}, {'end': 12675.852, 'text': 'This video gives formulas for sine of two theta and cosine of two theta.', 'start': 12670.029, 'duration': 5.823}, {'end': 12684.667, 'text': 'Please pause the video for a moment and see if you think this equation sine of 2 theta equals 2 sine theta is true or false.', 'start': 12677.401, 'duration': 7.266}, {'end': 12691.211, 'text': 'Remember that true means always true for all values of theta, where false means sometimes or always false.', 'start': 12685.327, 'duration': 5.884}], 'summary': 'Geometric proof of angle sum formulas and sine/cosine formulas provided in the video.', 'duration': 27.425, 'max_score': 12663.786, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/eI4an8aSsgw/pics/eI4an8aSsgw12663786.jpg'}, {'end': 13147.889, 'src': 'heatmap', 'start': 12925.781, 'weight': 0, 'content': [{'end': 12934.046, 'text': 'So that gives me cosine of two theta is cosine squared theta minus the quantity one minus cosine squared theta.', 'start': 12925.781, 'duration': 8.265}, {'end': 12943.332, 'text': 'That simplifies to two cosine squared theta minus one after distributing the negative sign and combining like terms.', 'start': 12935.207, 'duration': 8.125}, {'end': 12948.455, 'text': 'So I have one double angle formula for sine of two theta.', 'start': 12945.233, 'duration': 3.222}, {'end': 12952.344, 'text': 'And I have three versions of the double angle formula for cosine of two theta.', 'start': 12948.942, 'duration': 3.402}, {'end': 12955.507, 'text': "Now let's use these formulas in some examples.", 'start': 12953.325, 'duration': 2.182}, {'end': 12963.412, 'text': "Let's find the cosine of two theta, if we know that cosine theta is negative one over root 10.", 'start': 12957.028, 'duration': 6.384}, {'end': 12966.994, 'text': 'And theta terminates in quadrant three.', 'start': 12963.412, 'duration': 3.582}, {'end': 12971.858, 'text': 'we have a choice of three formulas for cosine of two theta.', 'start': 12966.994, 'duration': 4.864}, {'end': 12976.481, 'text': "I'm going to choose the second one because it only involves cosine of theta on the right side.", 'start': 12971.858, 'duration': 4.623}, {'end': 12979.568, 'text': 'and I already know my value for cosine theta.', 'start': 12976.926, 'duration': 2.642}, {'end': 12985.811, 'text': "Of course, I could use one of the other ones, but then I'd have to work out the value of sine theta.", 'start': 12979.588, 'duration': 6.223}, {'end': 12989.673, 'text': 'So, plugging in, I get cosine of two.', 'start': 12986.952, 'duration': 2.721}, {'end': 13002.601, 'text': 'theta is twice negative one over root 10 squared minus one, which simplifies to two tenths minus one, or negative eight tenths, negative four fifths.', 'start': 12989.673, 'duration': 12.928}, {'end': 13010.912, 'text': "Finally, let's solve the equation 2 cosine x plus sine of 2x equals 0.", 'start': 13003.968, 'duration': 6.944}, {'end': 13018.937, 'text': 'What makes this equation tricky is that one of the trig functions has the argument of just x, but the other trig function has the argument of 2x.', 'start': 13010.912, 'duration': 8.025}, {'end': 13023.54, 'text': 'So I want to use my double angle formula to rewrite sine of 2x.', 'start': 13019.458, 'duration': 4.082}, {'end': 13032.806, 'text': "I'll copy down the 2 cosine x, and now sine of 2x is equal to 2 sine x cosine x.", 'start': 13025.401, 'duration': 7.405}, {'end': 13043.689, 'text': 'At this point, I see a way to factor my equation, I can factor out a two cosine x from both of these two terms.', 'start': 13034.525, 'duration': 9.164}, {'end': 13053.254, 'text': 'That gives me one plus sine x, and the product there is equal to zero.', 'start': 13045.91, 'duration': 7.344}, {'end': 13061.178, 'text': 'That means that either two cosine x is equal to zero, or one plus sine x is equal to zero.', 'start': 13054.134, 'duration': 7.044}, {'end': 13066.955, 'text': 'that simplifies to cosine x equals zero, or sine x is negative one.', 'start': 13062.213, 'duration': 4.742}, {'end': 13085, 'text': 'Using my unit circle, I see that cosine of x is zero at pi over two and three pi over two, while sine of x is negative one at three pi over two.', 'start': 13068.375, 'duration': 16.625}, {'end': 13097.295, 'text': "there's some redundancy here, but my solution set is going to be pi over, plus multiples of two pi and three pi over two plus multiples of two pi.", 'start': 13085, 'duration': 12.295}, {'end': 13104.998, 'text': 'This video proved the double angle formulas sine of two theta is two sine theta cosine theta.', 'start': 13098.815, 'duration': 6.183}, {'end': 13110.601, 'text': 'And cosine of two theta is cosine squared theta minus sine squared theta.', 'start': 13105.899, 'duration': 4.702}, {'end': 13117.824, 'text': 'It also proved two alternate versions of the equation for cosine of two theta.', 'start': 13112.001, 'duration': 5.823}, {'end': 13128.361, 'text': 'This video is about the half angle formulas for computing cosine of theta over two and sine of theta over two.', 'start': 13120.097, 'duration': 8.264}, {'end': 13139.425, 'text': 'Cosine of theta over two is either a plus or minus the square root of one plus cosine theta all over two.', 'start': 13130.021, 'duration': 9.404}, {'end': 13147.889, 'text': 'To figure out whether to use plus or minus, you need to know something about what quadrant the angle theta over two is.', 'start': 13140.966, 'duration': 6.923}], 'summary': 'Exploring double angle and half angle formulas for trigonometric functions, solving equations using these formulas.', 'duration': 26.563, 'max_score': 12925.781, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/eI4an8aSsgw/pics/eI4an8aSsgw12925781.jpg'}, {'end': 13139.425, 'src': 'embed', 'start': 13105.899, 'weight': 1, 'content': [{'end': 13110.601, 'text': 'And cosine of two theta is cosine squared theta minus sine squared theta.', 'start': 13105.899, 'duration': 4.702}, {'end': 13117.824, 'text': 'It also proved two alternate versions of the equation for cosine of two theta.', 'start': 13112.001, 'duration': 5.823}, {'end': 13128.361, 'text': 'This video is about the half angle formulas for computing cosine of theta over two and sine of theta over two.', 'start': 13120.097, 'duration': 8.264}, {'end': 13139.425, 'text': 'Cosine of theta over two is either a plus or minus the square root of one plus cosine theta all over two.', 'start': 13130.021, 'duration': 9.404}], 'summary': 'The video covers half angle formulas for computing cosine and sine, including alternate versions of the equation.', 'duration': 33.526, 'max_score': 13105.899, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/eI4an8aSsgw/pics/eI4an8aSsgw13105899.jpg'}], 'start': 12468.671, 'title': 'Trigonometric identities and solving triangles', 'summary': 'Covers geometric proof of angle sum formulas, double angle formulas, half angle formulas, and solving right triangles through trigonometric functions, with quantifiable results in specific examples.', 'chapters': [{'end': 12727.027, 'start': 12468.671, 'title': 'Geometric proof of angle sum formulas', 'summary': 'Explains a geometric proof of the angle sum formulas, deriving side lengths in right triangles and discussing the truth of the equation sine of 2 theta equals 2 sine theta.', 'duration': 258.356, 'highlights': ['The chapter explains a geometric proof of the angle sum formulas', 'Deriving side lengths in right triangles', 'Discussion on the truth of the equation sine of 2 theta equals 2 sine theta']}, {'end': 13201.134, 'start': 12728.448, 'title': 'Double angle formulas and half angle formulas', 'summary': 'Discusses the derivation and application of double angle formulas for sine and cosine, including examples solving for cosine of two theta and solving equations involving sine of 2x. it also introduces the half angle formulas for computing cosine and sine of theta over two, emphasizing the use of plus or minus signs based on the quadrant of the angle.', 'duration': 472.686, 'highlights': ['The chapter derives and explains the double angle formulas: sine of two theta is two sine theta cosine theta and cosine of two theta is cosine squared theta minus sine squared theta, providing insight into their origins from the angle sum formula and the Pythagorean identity, as well as presenting alternate versions of the cosine of two theta formula.', 'Examples are provided, including solving for cosine of two theta when cosine theta is known, and solving equations involving sine of 2x using the double angle formula to rewrite sine of 2x for factorization and solution.', 'The chapter introduces the half angle formulas for cosine and sine of theta over two, highlighting the use of plus or minus signs based on the quadrant of the angle and explaining the reasoning behind the formulas with reference to the positivity of the square root expressions.']}, {'end': 13489.93, 'start': 13201.934, 'title': 'Half angle formulas in trigonometry', 'summary': 'Discusses the derivation and application of half angle formulas in trigonometry, providing a step-by-step process to find the exact values of cosine theta over two and sine theta over two using a specific example with quantifiable results.', 'duration': 287.996, 'highlights': ['Derivation of the half angle formulas', 'Application of the half angle formulas in an example', 'Quantifiable results from the example']}, {'end': 13844.716, 'start': 13491.152, 'title': 'Solving right triangle', 'summary': 'Explains how to solve a right triangle, using the example of finding angles and side lengths given partial information, through trigonometric functions and the pythagorean theorem, allowing the determination of all angles and side lengths of the triangle.', 'duration': 353.564, 'highlights': ['We used facts like tangent of an angle being opposite over adjacent and similar facts about sine and cosine.', "To find the measure of angle A, let's use the fact that the measures of the three angles of a triangle add up to 180 degrees.", 'We could use the fact that tan of 49 degrees, which is opposite over adjacent is B over 23. So B is 23 times tan 49 degrees.']}], 'duration': 1376.045, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/eI4an8aSsgw/pics/eI4an8aSsgw12468671.jpg', 'highlights': ['The chapter derives and explains the double angle formulas: sine of two theta is two sine theta cosine theta and cosine of two theta is cosine squared theta minus sine squared theta, providing insight into their origins from the angle sum formula and the Pythagorean identity, as well as presenting alternate versions of the cosine of two theta formula.', 'The chapter introduces the half angle formulas for cosine and sine of theta over two, highlighting the use of plus or minus signs based on the quadrant of the angle and explaining the reasoning behind the formulas with reference to the positivity of the square root expressions.', 'The chapter explains a geometric proof of the angle sum formulas', 'Examples are provided, including solving for cosine of two theta when cosine theta is known, and solving equations involving sine of 2x using the double angle formula to rewrite sine of 2x for factorization and solution.']}, {'end': 15452.738, 'segs': [{'end': 13894.148, 'src': 'embed', 'start': 13844.716, 'weight': 3, 'content': [{'end': 13847.557, 'text': 'Notice that we use many of the same ideas as in the previous problem.', 'start': 13844.716, 'duration': 2.841}, {'end': 13859.579, 'text': 'For example, the fact that the sum of the angles is 180, the Pythagorean theorem, and the trig functions, like tan, sine and cosine.', 'start': 13848.017, 'duration': 11.562}, {'end': 13866.441, 'text': 'We also use the inverse trig functions to get from an equation like this one to the angle.', 'start': 13860.34, 'duration': 6.101}, {'end': 13875.704, 'text': "This video showed how it's possible to find the lengths of all the sides of a right triangle and the measures of all the angles given partial information.", 'start': 13868.342, 'duration': 7.362}, {'end': 13882.705, 'text': 'For example, the measure of one angle and one side or from two sides.', 'start': 13876.744, 'duration': 5.961}, {'end': 13894.148, 'text': 'Recall that solving a triangle means finding all the lengths of the sides and all the measures of the angles from partial information.', 'start': 13885.506, 'duration': 8.642}], 'summary': 'Using trigonometric functions and the pythagorean theorem, we can find triangle lengths and angles from partial information.', 'duration': 49.432, 'max_score': 13844.716, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/eI4an8aSsgw/pics/eI4an8aSsgw13844716.jpg'}, {'end': 13944.845, 'src': 'embed', 'start': 13919.597, 'weight': 0, 'content': [{'end': 13927.079, 'text': 'I like to think of the law of cosines as a generalization of the Pythagorean theorem to triangles that are not necessarily right triangles.', 'start': 13919.597, 'duration': 7.482}, {'end': 13938.883, 'text': 'Loosely speaking, the law of cosines says that c squared is equal to a squared plus b squared plus a correction factor.', 'start': 13928.58, 'duration': 10.303}, {'end': 13944.845, 'text': "where the correction factor depends on the size of the angle, that's opposite to side c.", 'start': 13938.883, 'duration': 5.962}], 'summary': 'The law of cosines generalizes the pythagorean theorem to non-right triangles by adding a correction factor based on the size of the opposite angle.', 'duration': 25.248, 'max_score': 13919.597, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/eI4an8aSsgw/pics/eI4an8aSsgw13919597.jpg'}, {'end': 14499.09, 'src': 'embed', 'start': 14475.771, 'weight': 1, 'content': [{'end': 14483.516, 'text': "The law of cosines can be thought of as the Pythagorean theorem with a correction factor to account for triangles that aren't right triangles.", 'start': 14475.771, 'duration': 7.745}, {'end': 14492.843, 'text': 'Recall that solving a right triangle means finding all the lengths of the sides and all the measures of the angles from partial information.', 'start': 14485.638, 'duration': 7.205}, {'end': 14499.09, 'text': 'The law of cosines is a tool that can help us solve triangles that are not necessarily right triangles.', 'start': 14493.888, 'duration': 5.202}], 'summary': 'Law of cosines extends pythagorean theorem to solve non-right triangles.', 'duration': 23.319, 'max_score': 14475.771, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/eI4an8aSsgw/pics/eI4an8aSsgw14475771.jpg'}, {'end': 14625.482, 'src': 'embed', 'start': 14592.262, 'weight': 2, 'content': [{'end': 14595.704, 'text': "So I'll write it as a squared plus b squared plus a little bit.", 'start': 14592.262, 'duration': 3.442}, {'end': 14601.317, 'text': 'The Law of Cosines says exactly what this little correction factor is.', 'start': 14597.216, 'duration': 4.101}, {'end': 14611.559, 'text': 'It says that for any triangle with sides AB and C and angle capital C opposite to side C, C squared is equal to A squared plus,', 'start': 14602.017, 'duration': 9.542}, {'end': 14617.88, 'text': 'B squared minus 2AB cosine of angle C.', 'start': 14611.559, 'duration': 6.321}, {'end': 14625.482, 'text': 'Notice that if the angle C is less than 90 degrees, cosine of C is going to be a positive number.', 'start': 14617.88, 'duration': 7.602}], 'summary': 'Law of cosines gives c squared = a squared + b squared - 2ab cosine of angle c.', 'duration': 33.22, 'max_score': 14592.262, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/eI4an8aSsgw/pics/eI4an8aSsgw14592262.jpg'}, {'end': 15028.314, 'src': 'embed', 'start': 14995.697, 'weight': 5, 'content': [{'end': 15000.861, 'text': "and it can also be used in the situation where we know two sides and an angle that's not between them.", 'start': 14995.697, 'duration': 5.164}, {'end': 15017.251, 'text': 'The Law of Sines says that for a triangle with angles A, B and C opposite to sides lowercase a, b and c respectively, the sine of angle A,', 'start': 15002.362, 'duration': 14.889}, {'end': 15028.314, 'text': 'divided by the length of side A, is equal to the sine of B over B, which is also equal to the sine of C over C.', 'start': 15017.251, 'duration': 11.063}], 'summary': 'The law of sines relates angles and sides in a triangle.', 'duration': 32.617, 'max_score': 14995.697, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/eI4an8aSsgw/pics/eI4an8aSsgw14995697.jpg'}, {'end': 15399.017, 'src': 'embed', 'start': 15371.459, 'weight': 8, 'content': [{'end': 15380.565, 'text': 'To get side length C, we finished solving for the two possible triangles with these given side lengths and angle.', 'start': 15371.459, 'duration': 9.106}, {'end': 15387.014, 'text': 'In this video, we use the law of sines to solve a triangle in two different situations.', 'start': 15382.273, 'duration': 4.741}, {'end': 15391.415, 'text': 'In the first situation, we had two angles and the side between them.', 'start': 15387.534, 'duration': 3.881}, {'end': 15399.017, 'text': "So that's an A s a triangle for angle side angle.", 'start': 15391.915, 'duration': 7.102}], 'summary': 'Using the law of sines to solve triangles in two different situations.', 'duration': 27.558, 'max_score': 15371.459, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/eI4an8aSsgw/pics/eI4an8aSsgw15371459.jpg'}], 'start': 13844.716, 'title': 'Solving triangles', 'summary': 'Explains methods for solving right triangles using concepts like the sum of angles, pythagorean theorem, and trig functions, discusses the law of cosines for non-right triangles, and demonstrates the application of the law of cosines to find side lengths and angles, resulting in specific angle measurements of 60 degrees, 87.79 degrees, and 32.20 degrees.', 'chapters': [{'end': 13894.148, 'start': 13844.716, 'title': 'Solving right triangles', 'summary': 'Explains how to find the lengths of all the sides and measures of all the angles of a right triangle using partial information, leveraging concepts like the sum of angles, pythagorean theorem, and trig functions.', 'duration': 49.432, 'highlights': ['The chapter explains how to find the lengths of all the sides and measures of all the angles of a right triangle using partial information', 'Leveraging concepts like the sum of angles, Pythagorean theorem, and trig functions']}, {'end': 14388.156, 'start': 13895.305, 'title': 'Law of cosines in triangles', 'summary': 'Discusses the law of cosines as a generalization of the pythagorean theorem to solve non-right triangles, illustrating the relationship between side lengths and angles and how to apply the law of cosines to find side lengths and angles in triangles.', 'duration': 492.851, 'highlights': ['The law of cosines generalizes the Pythagorean theorem for non-right triangles', 'Relationship between angle size and side length in the law of cosines', 'Versatility of the law of cosines in triangle problem-solving']}, {'end': 14954.991, 'start': 14389.638, 'title': 'Solving triangles with law of cosines', 'summary': 'Demonstrates the application of the law of cosines to solve triangles, with detailed steps of finding side lengths and angles using the law of cosines, resulting in angles of 60 degrees, 87.79 degrees, and 32.20 degrees adding up to almost exactly 180 degrees.', 'duration': 565.353, 'highlights': ["The law of cosines can be thought of as the Pythagorean theorem with a correction factor to account for triangles that aren't right triangles.", 'The law of cosines states that for any triangle with sides AB and C and angle capital C opposite to side C, C squared is equal to A squared plus B squared minus 2AB cosine of angle C.', 'Solving a right triangle means finding all the lengths of the sides and all the measures of the angles from partial information.']}, {'end': 15452.738, 'start': 14955.932, 'title': 'Law of sines and cosines', 'summary': 'Discusses the application of the law of sines and cosines to solve triangles, demonstrating how to find angle measurements and side lengths using sine and cosine functions, with an emphasis on dealing with ambiguous cases.', 'duration': 496.806, 'highlights': ["The law of cosines can be thought of as the Pythagorean theorem with an adjustment factor to account for triangles that aren't right triangles.", 'The Law of Sines says that for a triangle with angles A, B and C opposite to sides lowercase a, b and c respectively, the sine of angle A, divided by the length of side A, is equal to the sine of B over B, which is also equal to the sine of C over C.', "In this video, we use the law of sines to solve a triangle in two different situations. In the first situation, we had two angles and the side between them. So that's an A s a triangle for angle side angle."]}], 'duration': 1608.022, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/eI4an8aSsgw/pics/eI4an8aSsgw13844716.jpg', 'highlights': ['The law of cosines generalizes the Pythagorean theorem for non-right triangles', 'The law of cosines can be thought of as the Pythagorean theorem with a correction factor', 'The law of cosines states that for any triangle with sides AB and C and angle capital C opposite to side C, C squared is equal to A squared plus B squared minus 2AB cosine of angle C', 'The chapter explains how to find the lengths of all the sides and measures of all the angles of a right triangle using partial information', 'Leveraging concepts like the sum of angles, Pythagorean theorem, and trig functions', 'The Law of Sines says that for a triangle with angles A, B and C opposite to sides lowercase a, b and c respectively, the sine of angle A, divided by the length of side A, is equal to the sine of B over B, which is also equal to the sine of C over C', 'Versatility of the law of cosines in triangle problem-solving', 'Solving a right triangle means finding all the lengths of the sides and all the measures of the angles from partial information', "In this video, we use the law of sines to solve a triangle in two different situations. In the first situation, we had two angles and the side between them. So that's an A s a triangle for angle side angle"]}, {'end': 17653.853, 'segs': [{'end': 15663.256, 'src': 'heatmap', 'start': 15454.605, 'weight': 0, 'content': [{'end': 15460.608, 'text': 'This video describes some of the features of a parabola, its vertex, focus and directrix.', 'start': 15454.605, 'duration': 6.003}, {'end': 15473.755, 'text': 'I want to start by finding the equation of all points that are equidistant the same distance from a point zero p on the y axis and the horizontal line.', 'start': 15462.129, 'duration': 11.626}, {'end': 15476.316, 'text': 'y equals negative p.', 'start': 15473.755, 'duration': 2.561}, {'end': 15479.518, 'text': "I'm assuming here that p is positive for starters.", 'start': 15476.316, 'duration': 3.202}, {'end': 15486.122, 'text': 'Certainly, the origin will be among those points, as it has a distance of p from the point.', 'start': 15481.047, 'duration': 5.075}, {'end': 15488.473, 'text': 'and P from the line.', 'start': 15486.972, 'duration': 1.501}, {'end': 15495.078, 'text': 'But the other points on the x axis will be closer to the line than they are to the point.', 'start': 15489.634, 'duration': 5.444}, {'end': 15503.364, 'text': "So, if I want the set of points that are the same distance from the point in the line, that's going to be a curve that curves upwards,", 'start': 15495.598, 'duration': 7.766}, {'end': 15504.184, 'text': 'something like this', 'start': 15503.364, 'duration': 0.82}, {'end': 15511.97, 'text': 'So for example, a point out here will be the same distance from the point as it is from the line.', 'start': 15505.425, 'duration': 6.545}, {'end': 15518.933, 'text': 'So our intuition is suggesting that this set of points should be the shape of a parabola.', 'start': 15513.43, 'duration': 5.503}, {'end': 15521.495, 'text': "Let's confirm this with some algebra.", 'start': 15519.474, 'duration': 2.021}, {'end': 15533.062, 'text': 'If we take an arbitrary point with coordinates x y, its distance from the point 0, p is given by the distance.', 'start': 15523.056, 'duration': 10.006}, {'end': 15537.365, 'text': 'formula x minus 0 squared, plus y minus p squared.', 'start': 15533.062, 'duration': 4.303}, {'end': 15547.276, 'text': 'Its distance from the line y equals negative p is just given by its difference in y coordinates.', 'start': 15538.672, 'duration': 8.604}, {'end': 15556.9, 'text': "So that's going to be y minus negative p, or y plus p.", 'start': 15548.056, 'duration': 8.844}, {'end': 15561.642, 'text': 'Let me set these two quantities equal to each other and simplify.', 'start': 15556.9, 'duration': 4.742}, {'end': 15563.543, 'text': 'I can square both sides.', 'start': 15562.362, 'duration': 1.181}, {'end': 15573.065, 'text': 'to get on the left x minus zero squared with the same thing as x squared plus y minus p squared equals y plus p squared.', 'start': 15564.861, 'duration': 8.204}, {'end': 15588.813, 'text': "Now I'll distribute out that gives me x squared plus y squared minus two p y plus p squared equals y squared plus two p y plus p squared.", 'start': 15573.645, 'duration': 15.168}, {'end': 15593.645, 'text': 'The p squareds cancel out, as do the y squareds.', 'start': 15590.339, 'duration': 3.306}, {'end': 15602.141, 'text': "And I'm left with x squared equals 4py after moving this negative 2py to the other side.", 'start': 15594.306, 'duration': 7.835}, {'end': 15609.378, 'text': 'I could also write this as y equals one over four p x squared.', 'start': 15603.874, 'duration': 5.504}, {'end': 15613.081, 'text': 'And you might recognize this as the standard parabola.', 'start': 15609.658, 'duration': 3.423}, {'end': 15621.867, 'text': 'y equals x squared, transformed by vertically stretched stretching or shrinking it by this factor of one over four p.', 'start': 15613.081, 'duration': 8.786}, {'end': 15625.509, 'text': 'you might recall that this lowest point of the parabola is called the vertex.', 'start': 15621.867, 'duration': 3.642}, {'end': 15629.072, 'text': 'This point here is called its focus.', 'start': 15626.61, 'duration': 2.462}, {'end': 15633.642, 'text': 'and the line here is called the directrix.', 'start': 15630.72, 'duration': 2.922}, {'end': 15647.554, 'text': 'Notice that the number p in this equation represents the distance between the vertex and the focus and also represents the distance between the vertex and the directrix line.', 'start': 15636.385, 'duration': 11.169}, {'end': 15657.655, 'text': "For this reason, if you're interested in the focus and the vertex, this form of the equation might be more useful than the equivalent form.", 'start': 15649.093, 'duration': 8.562}, {'end': 15663.256, 'text': 'something like y equals ax squared, where that distance, p, is more hidden.', 'start': 15657.655, 'duration': 5.601}], 'summary': "The parabola's equation for points equidistant from a point and a line is y = 1/(4p)x^2, where p represents the distance between the vertex, focus, and directrix.", 'duration': 26.871, 'max_score': 15454.605, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/eI4an8aSsgw/pics/eI4an8aSsgw15454605.jpg'}, {'end': 16429.411, 'src': 'heatmap', 'start': 16229.112, 'weight': 0.809, 'content': [{'end': 16233.977, 'text': 'The line segment that cuts through the middle of the ellipse in the shorter direction is called the minor axis.', 'start': 16229.112, 'duration': 4.865}, {'end': 16241.765, 'text': 'And the two points at the tips of the ellipse where the major axis actually touches the ellipse are called the vertices.', 'start': 16235.418, 'duration': 6.347}, {'end': 16246.791, 'text': 'An ellipse could be elongated in any direction,', 'start': 16244.09, 'duration': 2.701}, {'end': 16252.474, 'text': "but we'll only consider ellipses that are elongated either in the horizontal direction or the vertical direction.", 'start': 16246.791, 'duration': 5.683}, {'end': 16269.261, 'text': "Let's find the equation of an ellipse whose foci are at negative c0 and c0 and whose vertices are at negative a0 and a0.", 'start': 16253.414, 'duration': 15.847}, {'end': 16273.523, 'text': "This will be an ellipse that's elongated in the horizontal direction.", 'start': 16270.262, 'duration': 3.261}, {'end': 16287.004, 'text': 'For any point x, y on the ellipse, the sum of the distance from x, y to the first focus plus its distance to the second focus has to be some constant.', 'start': 16274.997, 'duration': 12.007}, {'end': 16292.047, 'text': 'And in fact, it turns out that constant has to equal 2a.', 'start': 16287.864, 'duration': 4.183}, {'end': 16293.347, 'text': "I'll show you why.", 'start': 16292.607, 'duration': 0.74}, {'end': 16309.023, 'text': 'If you look at this point right here on the far right tip of the ellipse, its distance from the first focus is going to be to C plus a minus C,', 'start': 16294.808, 'duration': 14.215}, {'end': 16313.629, 'text': 'and its distance from the second focus is just going to be a minus C.', 'start': 16309.023, 'duration': 4.606}, {'end': 16324.529, 'text': 'If you add up C plus C plus a minus C plus a minus C, you get exactly to a So let me write out my formulas for distances.', 'start': 16313.629, 'duration': 10.9}, {'end': 16330.733, 'text': 'The distance from a point xy to the first focus, the point negative c0,', 'start': 16325.19, 'duration': 5.543}, {'end': 16337.657, 'text': 'is going to be the square root of x minus negative c squared plus y minus 0 squared.', 'start': 16330.733, 'duration': 6.924}, {'end': 16348.463, 'text': 'And the distance from xy to the second focus, the point c0, is going to be the square root of x minus c squared plus y minus 0 squared.', 'start': 16338.557, 'duration': 9.906}, {'end': 16350.825, 'text': 'That sum needs to add up to 2a.', 'start': 16349.184, 'duration': 1.641}, {'end': 16355.053, 'text': 'After a fair amount of algebra.', 'start': 16353.132, 'duration': 1.921}, {'end': 16366.581, 'text': 'this simplifies to the expression a squared minus c squared x squared plus a squared y squared equals a squared times a squared minus c squared.', 'start': 16355.053, 'duration': 11.528}, {'end': 16374.984, 'text': 'If we let b squared equal a squared minus c squared, then we can rewrite this as b squared.', 'start': 16368.658, 'duration': 6.326}, {'end': 16376.745, 'text': 'x squared plus a squared.', 'start': 16374.984, 'duration': 1.761}, {'end': 16383.631, 'text': 'y squared equals a squared b squared and dividing both sides by a squared b squared.', 'start': 16376.745, 'duration': 6.886}, {'end': 16393.069, 'text': 'this gives us the form of the equation for the ellipse x squared over a squared plus y squared over b squared equals 1..', 'start': 16383.631, 'duration': 9.438}, {'end': 16400.375, 'text': 'Since we got b squared as a squared minus c squared, it follows that b is less than a.', 'start': 16393.069, 'duration': 7.306}, {'end': 16404.019, 'text': 'In fact, b is half the length of the minor axis.', 'start': 16400.375, 'duration': 3.644}, {'end': 16413.347, 'text': 'In other words, this point right here is the point zero b, and this point is the point zero negative b.', 'start': 16404.399, 'duration': 8.948}, {'end': 16416.93, 'text': "To see why this is true, let's draw these two right triangles.", 'start': 16413.347, 'duration': 3.583}, {'end': 16423.129, 'text': "The sum of the side lengths that I've drawn has to be the total length of the string.", 'start': 16418.427, 'duration': 4.702}, {'end': 16429.411, 'text': 'So that equals to a, which means each of these side lengths must be a.', 'start': 16423.569, 'duration': 5.842}], 'summary': 'Equation of an ellipse with foci and vertices, distance sum, and final form', 'duration': 200.299, 'max_score': 16229.112, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/eI4an8aSsgw/pics/eI4an8aSsgw16229112.jpg'}, {'end': 16313.629, 'src': 'embed', 'start': 16253.414, 'weight': 1, 'content': [{'end': 16269.261, 'text': "Let's find the equation of an ellipse whose foci are at negative c0 and c0 and whose vertices are at negative a0 and a0.", 'start': 16253.414, 'duration': 15.847}, {'end': 16273.523, 'text': "This will be an ellipse that's elongated in the horizontal direction.", 'start': 16270.262, 'duration': 3.261}, {'end': 16287.004, 'text': 'For any point x, y on the ellipse, the sum of the distance from x, y to the first focus plus its distance to the second focus has to be some constant.', 'start': 16274.997, 'duration': 12.007}, {'end': 16292.047, 'text': 'And in fact, it turns out that constant has to equal 2a.', 'start': 16287.864, 'duration': 4.183}, {'end': 16293.347, 'text': "I'll show you why.", 'start': 16292.607, 'duration': 0.74}, {'end': 16309.023, 'text': 'If you look at this point right here on the far right tip of the ellipse, its distance from the first focus is going to be to C plus a minus C,', 'start': 16294.808, 'duration': 14.215}, {'end': 16313.629, 'text': 'and its distance from the second focus is just going to be a minus C.', 'start': 16309.023, 'duration': 4.606}], 'summary': 'Equation of ellipse with foci at -c0 and c0, vertices at -a0 and a0, sum of distances to foci equals 2a.', 'duration': 60.215, 'max_score': 16253.414, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/eI4an8aSsgw/pics/eI4an8aSsgw16253414.jpg'}, {'end': 16484.877, 'src': 'embed', 'start': 16445.407, 'weight': 2, 'content': [{'end': 16454.376, 'text': 'To summarize, for a number a bigger than b, the equation x, squared over a squared, plus y, squared over b squared equals one,', 'start': 16445.407, 'duration': 8.969}, {'end': 16462.784, 'text': "represents an ellipse that's elongated in the horizontal direction.", 'start': 16454.376, 'duration': 8.408}, {'end': 16479.436, 'text': 'whose major axis terminates in the points minus a zero and a zero and whose minor axis terminates in the points zero b and zero minus b.', 'start': 16464.331, 'duration': 15.105}, {'end': 16484.877, 'text': 'Its foci are at the points negative c zero and c zero.', 'start': 16479.436, 'duration': 5.441}], 'summary': 'Ellipse equation: x^2/a^2 + y^2/b^2 = 1, major axis at −a 0 and a 0, minor axis at 0 −b and 0 b, foci at −c 0 and c 0.', 'duration': 39.47, 'max_score': 16445.407, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/eI4an8aSsgw/pics/eI4an8aSsgw16445407.jpg'}, {'end': 16853.666, 'src': 'embed', 'start': 16820.646, 'weight': 3, 'content': [{'end': 16831.108, 'text': 'This video detailed the anatomy of ellipse, including its major axis, its minor axis, its vertices, and its foci.', 'start': 16820.646, 'duration': 10.462}, {'end': 16835.389, 'text': 'This video is about hyperbolas.', 'start': 16833.429, 'duration': 1.96}, {'end': 16848.152, 'text': 'You might recall that an ellipse is the set of points x y, such that the sum of the distances between x,', 'start': 16838.09, 'duration': 10.062}, {'end': 16853.666, 'text': 'y and two fixed points called the foci is a constant.', 'start': 16848.152, 'duration': 5.514}], 'summary': 'Video details ellipse anatomy and its properties.', 'duration': 33.02, 'max_score': 16820.646, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/eI4an8aSsgw/pics/eI4an8aSsgw16820646.jpg'}], 'start': 15454.605, 'title': 'Equations of conic sections', 'summary': "Discusses the equations and features of parabolas, exploring variations when p is greater or less than 0, equations for points on the x-axis, and also covers ellipses, their features, and properties, as well as hyperbolas' anatomy, orientation, and key elements.", 'chapters': [{'end': 15823.883, 'start': 15454.605, 'title': 'Equation of a parabola and its features', 'summary': 'Discusses finding the equation of all points equidistant from a point and a line, resulting in the standard form of a parabola y = 1/(4p)x^2, where p signifies the distance between the vertex and the focus/directrix, and explores the variations when p is greater than or less than 0 as well as the equation for points equidistant from a point on the x-axis, yielding x = 1/(4p)y^2.', 'duration': 369.278, 'highlights': ['The standard form of the parabola is y = 1/(4p)x^2, where p represents the distance between the vertex and the focus/directrix.', "Exploration of variations when p is greater than or less than 0 and the resulting changes in the parabola's orientation and coefficient of y or x.", 'The equation for points equidistant from a point on the x-axis yields x = 1/(4p)y^2, signifying a parabola pointing sideways.']}, {'end': 16166.026, 'start': 15823.883, 'title': 'Equation of a parabola with vertex and focus', 'summary': 'Covers the equation of a parabola with a shifted vertex and focus, highlighting the transformation of the parabola equation, determining the direction of opening, and finding the specific equation of a parabola with given vertex and focus coordinates.', 'duration': 342.143, 'highlights': ["The equation y minus k squared equals four p x minus h and x minus h squared equals four p y minus k represents the vertex, with P's absolute value indicating the distance between the vertex and the focus and directrix, enabling the determination of the parabola's direction of opening and specific equation.", 'The transformation of the parabola equation involves shifting the vertex right by h and up by k, accomplished by x minus h and plus k, where the equations remain the same but with added h and k, facilitating the understanding of how the parabola shifts in different directions.', 'Determining the direction of opening for the parabola is based on the sign of p, with p greater than zero indicating the parabola opens right and up, and p less than zero indicating the parabola opens left and down, providing a clear guideline for understanding the orientation of the parabola.', 'The specific equation of a parabola with a vertex at two, four and focus at negative one, four is found using the formula y minus k squared equals four p x minus h, where the vertex and focus coordinates are used to determine the value of p and subsequently derive the equation of the parabola.']}, {'end': 16443.556, 'start': 16167.952, 'title': 'Features of an ellipse', 'summary': 'Explains the definition of an ellipse, its features including the major axis, minor axis, and equation, and the relationship between the foci, vertices, and constants, with detailed algebraic derivations and geometric explanations.', 'duration': 275.604, 'highlights': ['The equation of an ellipse with foci at negative c0 and c0 and vertices at negative a0 and a0 is derived as b squared. x squared plus a squared. y squared equals a squared b squared, and b is less than a, where b is half the length of the minor axis and a is the length of the semi-major axis.', 'The sum of the distances from a point on the ellipse to the two foci has to be a constant, which equals 2a, demonstrated through the algebraic simplification and the geometric explanation using right triangles to show the relationship between a, b, and c.', 'The major axis and the minor axis are defined, and the features of an ellipse, including the focuses, vertices, and constants, are explained with detailed geometric descriptions and algebraic derivations.']}, {'end': 16820.646, 'start': 16445.407, 'title': 'Ellipse and its properties', 'summary': 'Explains the equation of an ellipse, its properties, and how to shift its center, with a focus on the relationship between a, b, and c, and the calculation of the foci coordinates.', 'duration': 375.239, 'highlights': ['The equation x squared over a squared plus y squared over b squared equals one represents an ellipse elongated in the horizontal direction, with major axis terminating at points (-a, 0) and (a, 0) and minor axis terminating at points (0, b) and (0, -b). The foci are at points (-c, 0) and (c, 0), with the relationship b squared = a squared - c squared.', 'When a bigger than b and the roles of x and y are reversed, the ellipse is elongated in the vertical direction, with the major axis on the y-axis and the minor axis on the x-axis, and the foci are on the y-axis. Shifting the ellipse to an arbitrary point (h, k) requires adjusting the equation and center coordinates.', 'The detailed calculation of the foci coordinates for an ellipse elongated vertically, with the center at (h, k) = (4, -3), major axis length a = 6, and minor axis length b = 5, results in the equation x-4 squared over 5 squared plus y+3 squared over 6 squared equals 1 and the foci coordinates at (4, -3 ± √11).']}, {'end': 17653.853, 'start': 16820.646, 'title': 'Anatomy of hyperbolas', 'summary': 'Explains the anatomy of hyperbolas, including their major and minor axes, vertices, foci, and equations, highlighting the relationship between these elements and their orientation, with a focus on horizontal and vertical orientations and the corresponding equations. it also details how to find the center, vertices, foci, and asymptotes of a hyperbola, along with a numerical example.', 'duration': 833.207, 'highlights': ['The chapter explains the anatomy of hyperbolas, including their major and minor axes, vertices, and foci, and how they differ from ellipses.', 'It details the orientation of hyperbolas and their corresponding transverse axes, emphasizing horizontal and vertical orientations and their visualization.', 'The chapter provides the equation of a hyperbola with foci at negative c0 and c0 and vertices at negative a0 and a0, highlighting the relationship between a, b, and c in the equation.', 'It explains the equations of asymptotes for hyperbolas centered at the origin, detailing the relationship between a and b and the slopes of the asymptotes.', 'The chapter demonstrates how to handle hyperbolas centered at an arbitrary point, providing equations for vertices, foci, and asymptotes in such cases.', 'It concludes with a numerical example, illustrating the process of finding the center, vertices, foci, and asymptotes of a hyperbola with specific numerical values, emphasizing the use of a and b in determining orientation and distances.']}], 'duration': 2199.248, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/eI4an8aSsgw/pics/eI4an8aSsgw15454605.jpg', 'highlights': ['The standard form of the parabola is y = 1/(4p)x^2, where p represents the distance between the vertex and the focus/directrix.', 'The equation of an ellipse with foci at negative c0 and c0 and vertices at negative a0 and a0 is derived as b squared. x squared plus a squared. y squared equals a squared b squared, and b is less than a, where b is half the length of the minor axis and a is the length of the semi-major axis.', 'The equation x squared over a squared plus y squared over b squared equals one represents an ellipse elongated in the horizontal direction, with major axis terminating at points (-a, 0) and (a, 0) and minor axis terminating at points (0, b) and (0, -b). The foci are at points (-c, 0) and (c, 0), with the relationship b squared = a squared - c squared.', 'The chapter explains the anatomy of hyperbolas, including their major and minor axes, vertices, and foci, and how they differ from ellipses.', 'The sum of the distances from a point on the ellipse to the two foci has to be a constant, which equals 2a, demonstrated through the algebraic simplification and the geometric explanation using right triangles to show the relationship between a, b, and c.']}, {'end': 19321.029, 'segs': [{'end': 18171.043, 'src': 'embed', 'start': 18143.393, 'weight': 4, 'content': [{'end': 18155.998, 'text': "This is especially useful as a way to describe curves that don't satisfy the vertical line test and therefore can't be described traditionally as functions of y in terms of x.", 'start': 18143.393, 'duration': 12.605}, {'end': 18160.499, 'text': 'A Cartesian equation for a curve is an equation in terms of x and y only.', 'start': 18155.998, 'duration': 4.501}, {'end': 18168.822, 'text': 'Parametric equations for a curve give both x and y as functions of a third variable, usually t.', 'start': 18161.499, 'duration': 7.323}, {'end': 18171.043, 'text': 'The third variable is called the parameter.', 'start': 18168.822, 'duration': 2.221}], 'summary': 'Parametric equations describe non-functional curves using x, y, and a third variable.', 'duration': 27.65, 'max_score': 18143.393, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/eI4an8aSsgw/pics/eI4an8aSsgw18143393.jpg'}, {'end': 18507.346, 'src': 'embed', 'start': 18475.444, 'weight': 5, 'content': [{'end': 18479.747, 'text': "We've seen several examples where we went from parametric equations to Cartesian equations.", 'start': 18475.444, 'duration': 4.303}, {'end': 18485.572, 'text': "Now let's start with a Cartesian equation and rewrite it as a parametric equation.", 'start': 18480.508, 'duration': 5.064}, {'end': 18490.695, 'text': 'In this example, y is already given as a function of x.', 'start': 18486.953, 'duration': 3.742}, {'end': 18503.205, 'text': 'So an easy way to parametrize this curve is to just let x equal t and then y is equal to the square root of t squared minus t,', 'start': 18490.695, 'duration': 12.51}, {'end': 18507.346, 'text': 'substituting in t for x.', 'start': 18503.205, 'duration': 4.141}], 'summary': 'Demonstrated conversion from cartesian to parametric equations.', 'duration': 31.902, 'max_score': 18475.444, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/eI4an8aSsgw/pics/eI4an8aSsgw18475444.jpg'}, {'end': 18738.201, 'src': 'embed', 'start': 18626.294, 'weight': 6, 'content': [{'end': 18638.599, 'text': 'This gives me the parameterization x equals 6 cosine of t, y equals 5 sine of t, which is a handy way to describe an ellipse.', 'start': 18626.294, 'duration': 12.305}, {'end': 18649.444, 'text': "As a final example, let's describe a general circle of radius r and center hk.", 'start': 18641.98, 'duration': 7.464}, {'end': 18663.732, 'text': 'For any point x, y on the circle, we know that the distance from that point to the center of the circle is equal to r.', 'start': 18651.464, 'duration': 12.268}, {'end': 18676.941, 'text': 'So using the distance formula, we know that the square root of squared plus squared has to equal r.', 'start': 18663.732, 'duration': 13.209}, {'end': 18682.385, 'text': 'Squaring both sides, this gives us the equation for the circle in Cartesian coordinates.', 'start': 18676.941, 'duration': 5.444}, {'end': 18697.239, 'text': 'So, for example, if our circle has radius five and has center at the point negative 317, then its equation would be x minus negative three.', 'start': 18683.729, 'duration': 13.51}, {'end': 18705.757, 'text': "that's x plus three squared plus y minus 17 squared is equal to 25..", 'start': 18697.239, 'duration': 8.518}, {'end': 18714.44, 'text': 'One way to find the equation of a general circle in parametric equations is to start with the unit circle and work our way up.', 'start': 18705.757, 'duration': 8.683}, {'end': 18729.305, 'text': 'We know that the unit circle with radius 1 centered at the origin is given by the equation x equals cosine t and y equals sine t.', 'start': 18715.58, 'duration': 13.725}, {'end': 18738.201, 'text': 'If we want a circle of radius r centered around the origin instead, then we need to expand everything by a factor of r.', 'start': 18729.305, 'duration': 8.896}], 'summary': 'Parametric and cartesian equations describe circles and ellipses, with examples and equations provided.', 'duration': 111.907, 'max_score': 18626.294, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/eI4an8aSsgw/pics/eI4an8aSsgw18626294.jpg'}, {'end': 18807.361, 'src': 'embed', 'start': 18767.533, 'weight': 9, 'content': [{'end': 18785.298, 'text': 'We can write our same example circle in parametric equations as x equals 5 cosine t minus 3, y equals 5 sine t plus 17.', 'start': 18767.533, 'duration': 17.765}, {'end': 18793.556, 'text': 'In this video we translated back and forth in between Cartesian equations and and parametric equations,', 'start': 18785.298, 'duration': 8.258}, {'end': 18797.117, 'text': 'with a special emphasis on the equations for circles.', 'start': 18793.556, 'duration': 3.561}, {'end': 18807.361, 'text': 'This video is about the difference quotient and the average rate of change.', 'start': 18803.079, 'duration': 4.282}], 'summary': 'Video covers translating between cartesian and parametric equations for circles, with a focus on difference quotient and average rate of change.', 'duration': 39.828, 'max_score': 18767.533, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/eI4an8aSsgw/pics/eI4an8aSsgw18767533.jpg'}, {'end': 18978.659, 'src': 'embed', 'start': 18919.621, 'weight': 2, 'content': [{'end': 18923.942, 'text': 'So this average rate of change is the amount the tree grows in a certain time period.', 'start': 18919.621, 'duration': 4.321}, {'end': 18933.983, 'text': 'For example, if it grows 10 inches in two years, that would be 10 inches per two years, or five inches per year would be its average rate of change,', 'start': 18923.982, 'duration': 10.001}, {'end': 18935.064, 'text': 'its average rate of growth.', 'start': 18933.983, 'duration': 1.081}, {'end': 18942.248, 'text': "Let's compute the average rate of change for the function f of x equals square root of x on the interval from one to four.", 'start': 18936.625, 'duration': 5.623}, {'end': 18957.961, 'text': 'So the average rate of change is f of four minus f of one over four minus one, well, f of four is the square root of four of one square root of one.', 'start': 18944.269, 'duration': 13.692}, {'end': 18963.566, 'text': "So that's going to be two minus one over three or one third.", 'start': 18959.042, 'duration': 4.524}, {'end': 18978.659, 'text': "Instead of calling these two locations on the x axis, A and B, this time I'm going to call the first location just x and the second location x plus h.", 'start': 18965.508, 'duration': 13.151}], 'summary': 'Average rate of change is 1/3 for f(x) = √x from 1 to 4.', 'duration': 59.038, 'max_score': 18919.621, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/eI4an8aSsgw/pics/eI4an8aSsgw18919621.jpg'}, {'end': 19052.66, 'src': 'embed', 'start': 19008.88, 'weight': 0, 'content': [{'end': 19015.764, 'text': 'A difference quotient is simply the average rate of change using this x x plus h notation.', 'start': 19008.88, 'duration': 6.884}, {'end': 19031.552, 'text': 'So a difference quotient represents the average rate of change of a function f of x on the interval from x to x plus h.', 'start': 19017.004, 'duration': 14.548}, {'end': 19046.715, 'text': 'Equivalently. the difference quotient represents the slope of the secant line for the graph of y equals f of x between the points with coordinates x,', 'start': 19031.552, 'duration': 15.163}, {'end': 19052.66, 'text': 'f of x and x plus h, f of x plus h.', 'start': 19046.715, 'duration': 5.945}], 'summary': 'A difference quotient is the average rate of change using x + h notation.', 'duration': 43.78, 'max_score': 19008.88, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/eI4an8aSsgw/pics/eI4an8aSsgw19008880.jpg'}], 'start': 17655.965, 'title': 'Parametric equations and polar coordinates', 'summary': 'Introduces polar coordinates, parametric equations, and their conversions, covering curves, circles, and average rate of change with examples and computations.', 'chapters': [{'end': 18118.579, 'start': 17655.965, 'title': 'Polar coordinates and conversions', 'summary': 'Introduces the concept of polar coordinates, explaining how to plot points using r and theta, convert between polar and cartesian coordinates using trigonometric equations, and discusses the idea of parametric equations.', 'duration': 462.614, 'highlights': ['Polar coordinates introduce an alternative way of describing point location using radius r and angle theta, providing a different perspective from Cartesian coordinates.', 'Explains the process of plotting points in polar coordinates by illustrating how to interpret radius and angle values to determine point placement on the plane.', 'Discusses the conversion between polar and Cartesian coordinates using equations x = r*cos(theta), y = r*sin(theta), r^2 = x^2 + y^2, and tangent(theta) = y/x, providing a method for translating between the two coordinate systems.', 'Introduces the concept of parametric equations, expanding the understanding of coordinate systems and mathematical representations.']}, {'end': 18528.694, 'start': 18119.279, 'title': 'Parametric equations explained', 'summary': 'Covers the concept of parametric equations, providing examples of how to graph and convert them to cartesian equations, including a unit circle and a sideways parabola, with key points such as the parameter t representing time and the copycat parameterization method.', 'duration': 409.415, 'highlights': ["The concept of parametric equations and their use to describe curves that don't satisfy the vertical line test is explained, with x and y coordinates represented as functions of a third variable, usually time.", 'Examples of parametric equations are provided, including graphing the equations on an xy coordinate axis and converting them to Cartesian equations, such as a unit circle and a sideways parabola.', 'The concept of the parameter t as a representation of time is emphasized, with examples demonstrating the traversal of curves in both clockwise and counterclockwise directions as t varies.', "The 'copycat parameterization' method is introduced, explaining how to rewrite Cartesian equations as parametric equations by letting t equal x and y equal a function involving t."]}, {'end': 18766.193, 'start': 18528.694, 'title': 'Parameterizing curves and circles', 'summary': 'Discusses parameterization of curves using sine and cosine, providing the parameterization for an ellipse and the general equation for a circle in parametric and cartesian coordinates.', 'duration': 237.499, 'highlights': ['The parameterization x equals 6 cosine of t, y equals 5 sine of t describes an ellipse.', 'The general equation for a circle in parametric equations is derived by expanding the unit circle by a factor of r and adjusting the center coordinates.', 'The equation for the circle in Cartesian coordinates is illustrated using the example of a circle with a radius of five and a center at the point (-3, 17).']}, {'end': 19007.243, 'start': 18767.533, 'title': 'Parametric equations and average rate of change', 'summary': "Covers the translation between cartesian and parametric equations focusing on circles and explains the concept of average rate of change using the example of a tree's growth and computes the average rate of change for the function f(x) = sqrt(x) on the interval from one to four.", 'duration': 239.71, 'highlights': ['The chapter covers the translation between Cartesian and parametric equations focusing on circles', "The chapter explains the concept of average rate of change using the example of a tree's growth", 'The chapter computes the average rate of change for the function f(x) = sqrt(x) on the interval from one to four']}, {'end': 19321.029, 'start': 19008.88, 'title': 'Calculating difference quotient', 'summary': 'Discusses the concept of difference quotient as the average rate of change of a function, and then derives a formula for the difference quotient, which is f of x plus h minus f of x over h, highlighting the importance of this concept in calculus.', 'duration': 312.149, 'highlights': ['The difference quotient represents the average rate of change of a function f of x on the interval from x to x plus h, or equivalently, the slope of the secant line for the graph of y equals f of x between the points with coordinates x, f of x and x plus h, f of x plus h.', 'The formula for the difference quotient is f of x plus h minus f of x over h, which eventually becomes important in calculus when calculating a difference quotient for smaller and smaller values of h, leading to the derivative or slope of the function itself.', 'The process of computing and simplifying the difference quotient involves substituting x plus h into the function, simplifying the expression, canceling out like terms, and factoring out h to obtain the difference quotient of 4x plus 2h minus 1.', 'The concept of difference quotient is related to the average rate of change, calculated using the formula f of b minus f of a over b minus a, and the difference quotient formula f of x plus h minus f of x over h.']}], 'duration': 1665.064, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/eI4an8aSsgw/pics/eI4an8aSsgw17655965.jpg', 'highlights': ['The difference quotient represents the average rate of change of a function f of x on the interval from x to x plus h, or equivalently, the slope of the secant line for the graph of y equals f of x between the points with coordinates x, f of x and x plus h, f of x plus h.', 'The concept of difference quotient is related to the average rate of change, calculated using the formula f of b minus f of a over b minus a, and the difference quotient formula f of x plus h minus f of x over h.', "The chapter explains the concept of average rate of change using the example of a tree's growth", 'The chapter computes the average rate of change for the function f(x) = sqrt(x) on the interval from one to four', "The concept of parametric equations and their use to describe curves that don't satisfy the vertical line test is explained, with x and y coordinates represented as functions of a third variable, usually time.", "The 'copycat parameterization' method is introduced, explaining how to rewrite Cartesian equations as parametric equations by letting t equal x and y equal a function involving t.", 'The parameterization x equals 6 cosine of t, y equals 5 sine of t describes an ellipse.', 'The general equation for a circle in parametric equations is derived by expanding the unit circle by a factor of r and adjusting the center coordinates.', 'The equation for the circle in Cartesian coordinates is illustrated using the example of a circle with a radius of five and a center at the point (-3, 17).', 'Introduces polar coordinates, parametric equations, and their conversions, covering curves, circles, and average rate of change with examples and computations.']}], 'highlights': ['The function notation f(x) represents the output value of y, with f(2) resulting in 5 and f(5) resulting in 26.', 'The toolkit functions include linear, quadratic, cubic, square root, absolute value, exponential, and hyperbola functions, each with specific symmetries and shapes.', 'The circumference of a circle is given by the formula, the circumference equals two pi times the radius.', 'The chapter covers properties of sine and cosine functions, including the periodic property stating that the values of cosine and sine are periodic with period two pi, and the even-odd property stating that cosine is even while sine is odd.', 'The law of cosines generalizes the Pythagorean theorem for non-right triangles', 'The equation of an ellipse with foci at negative c0 and c0 and vertices at negative a0 and a0 is derived as b squared. x squared plus a squared. y squared equals a squared b squared, and b is less than a, where b is half the length of the minor axis and a is the length of the semi-major axis.', 'The concept of difference quotient is related to the average rate of change, calculated using the formula f of b minus f of a over b minus a, and the difference quotient formula f of x plus h minus f of x over h.', "The concept of parametric equations and their use to describe curves that don't satisfy the vertical line test is explained, with x and y coordinates represented as functions of a third variable, usually time.", 'Introduces polar coordinates, parametric equations, and their conversions, covering curves, circles, and average rate of change with examples and computations.', 'The chapter concludes with mnemonic for sum and difference formulas for sine and cosine.']}