1 00:00:00,012 --> 00:00:04,084 [music]. Once, we believe that sine and cosine are 2 00:00:04,084 --> 00:00:10,213 important functions, they're all about the connection between angles and lengths. 3 00:00:10,213 --> 00:00:14,465 Well, then we want to apply our usual calculus trick. 4 00:00:14,465 --> 00:00:17,602 What happens if I wiggle the input to sine and cosine? 5 00:00:17,602 --> 00:00:22,062 You might think about trigonometry as being something about right triangles. 6 00:00:22,062 --> 00:00:26,484 But, here, I've drawn a picture of a circle and you can rephrase all this stuff 7 00:00:26,484 --> 00:00:29,137 in terms of just geometry of the unit circle. 8 00:00:29,137 --> 00:00:32,830 So this is the unit circle. The length of this radius is 1 and I've 9 00:00:32,830 --> 00:00:35,827 got a right triangle here and this angle is theta. 10 00:00:35,828 --> 00:00:41,165 That means this base here has like cosine theta and the height of this triangle is 11 00:00:41,165 --> 00:00:44,698 sine theta. The coordinates then, or this point on the 12 00:00:44,698 --> 00:00:50,037 unit circle are cosine theta sine theta. Now, let's make that angle just a little 13 00:00:50,037 --> 00:00:53,691 bit bigger. Let's suppose that I make this angle just 14 00:00:53,691 --> 00:00:59,086 a big bigger, and instead of thinking about theta, I'm thinking about an angle 15 00:00:59,086 --> 00:01:04,266 of measure theta plus h and that means this angle in between has measure h. 16 00:01:04,266 --> 00:01:09,690 We'll do a little bit of geometry to figure out how far that point moved when I 17 00:01:09,690 --> 00:01:15,722 wiggled from theta to theta plus h. So let's think about this point here and 18 00:01:15,722 --> 00:01:21,660 let's draw the tangent line to the circle at that particular point. 19 00:01:21,660 --> 00:01:27,602 Well, I'm going to draw a little tiny triangle, which is heading in the 20 00:01:27,602 --> 00:01:34,664 direction of that of that tangent line. And, so that it's tangent to the circle, 21 00:01:34,664 --> 00:01:41,524 the hypotenuse of that little tiny right triangle here in red will be perpendicular 22 00:01:41,524 --> 00:01:45,947 to this line here. I can actually determine the angles in 23 00:01:45,947 --> 00:01:51,362 that little, tiny right triangle. So we go this big right triangle down 24 00:01:51,362 --> 00:01:57,342 here, and just because its a triangle, this angle theta plus this angle in here 25 00:01:57,342 --> 00:02:01,321 plus this right angle have to add up to 180 degrees. 26 00:02:01,321 --> 00:02:08,608 But I've also got a straight line here and that means that this top angle plus this 27 00:02:08,608 --> 00:02:14,687 right angle plus this mystery angle must also add up to 180 degrees. 28 00:02:14,687 --> 00:02:18,711 Well, this plus this plus this is 180 degrees. 29 00:02:18,711 --> 00:02:24,999 And this plus this, same right angle, plus this mystery angle add up to 180 degrees. 30 00:02:24,999 --> 00:02:31,303 That means that, that mystery angle and that little tiny right triangle must also 31 00:02:31,303 --> 00:02:34,823 be theta. I'd also like to know the length of the 32 00:02:34,823 --> 00:02:39,503 hypotenuse of the little tiny triangle. Radians save the day. 33 00:02:39,503 --> 00:02:42,782 How so? Well, I want to know the length of this 34 00:02:42,782 --> 00:02:45,824 little piece of arc. What else do I know? 35 00:02:45,824 --> 00:02:50,357 I know this is a unit circle. And I know this angle here is h in 36 00:02:50,357 --> 00:02:54,199 radians. And the definition of radiance means that 37 00:02:54,199 --> 00:02:57,431 this little length of arc here has length h. 38 00:02:57,431 --> 00:03:01,516 Now, let's put my tiny right triangle back there. 39 00:03:01,516 --> 00:03:06,718 I'm going to have the hypotenuse of that little right triangle also be h. 40 00:03:06,718 --> 00:03:11,235 It's going to be close enough, right, because this little piece of curved arch 41 00:03:11,235 --> 00:03:15,651 and this straight line are awfully close. Now that I know the length of the 42 00:03:15,651 --> 00:03:20,124 hypotenuse and the angles in that right triangle I can use sine and cosine to 43 00:03:20,124 --> 00:03:23,138 determine the side lengths of the other two sides. 44 00:03:23,138 --> 00:03:28,720 So I've got a little tiny right triangle, hypotenuse h, that angle there is theta, 45 00:03:28,720 --> 00:03:33,746 and that means that this vertical distance here is h times cosine theta. 46 00:03:33,746 --> 00:03:39,881 And the horizontal distance of that little tiny red triangle is h sine theta. 47 00:03:39,881 --> 00:03:43,425 Okay. So how much did wiggling from theta to 48 00:03:43,425 --> 00:03:47,241 theta plus h move the point around the circle? 49 00:03:47,241 --> 00:03:52,597 So the original point here from angle theta had coordinates cosine theta sine 50 00:03:52,597 --> 00:03:55,645 theta. And the point up here which I got when I 51 00:03:55,645 --> 00:04:01,057 wiggled theta up to theta plus h, that point has coordinates cosine theta plus h 52 00:04:01,057 --> 00:04:04,932 sine theta plus h. So how much did wiggling from theta to 53 00:04:04,932 --> 00:04:09,759 theta plus h move the point? Well, if I use this little right triangle 54 00:04:09,759 --> 00:04:15,624 as the approximation, sine theta increased by about H cosign data, and cosign data 55 00:04:15,624 --> 00:04:20,702 decreased by about h sine theta. We're now in a position to make a claim 56 00:04:20,702 --> 00:04:25,126 about the derivative. In other words, from the picture, we 57 00:04:25,126 --> 00:04:30,409 learned that sine theta plus h is about sine theta plus h cosine theta. 58 00:04:30,409 --> 00:04:34,780 And cosine theta plus h is about cosine theta minus h sine theta. 59 00:04:34,780 --> 00:04:39,052 And as a result, we can say something now about the derivatives. 60 00:04:39,052 --> 00:04:43,927 How does changing theta affect sine? Well, about a factor of cosine theta 61 00:04:43,927 --> 00:04:48,287 compared to the input. So the derivative of sine is cosine theta. 62 00:04:48,287 --> 00:04:53,830 And how does changing theta affect cosine? Well, about a factor of negative sine. 63 00:04:53,830 --> 00:04:56,453 So the derivative of cosine is negative sine. 64 00:04:56,453 --> 00:04:59,602 Maybe you don't find all this geometry convincing. 65 00:04:59,602 --> 00:05:04,207 Well, we could go back to the definition of derivative in terms of limits and 66 00:05:04,207 --> 00:05:09,357 calculate the derivative of sin directly. So if I want to calculate the derivative 67 00:05:09,357 --> 00:05:14,389 of sin using the limit definition of the derivative, well, the derivative of sine 68 00:05:14,389 --> 00:05:19,206 would be the limit as h approaches 0 of sine theta plus h minus sine theta over h. 69 00:05:19,206 --> 00:05:23,961 The trouble now, is that I've gotta somehow calculate sine theta plus h. 70 00:05:23,961 --> 00:05:27,983 How can I do that? Well, you might remember, there's an angle 71 00:05:27,983 --> 00:05:31,801 sum formula for sine. Sine of alpha plus beta is sine alpha 72 00:05:31,801 --> 00:05:37,242 cosine beta plus cosine alpha sine beta. If I use this, but replace alpha by theta 73 00:05:37,242 --> 00:05:42,432 and beta by h, I get this. Sine of theta plus h can be replaced by 74 00:05:42,432 --> 00:05:46,554 sine theta cosine h plus cosine theta sine h. 75 00:05:46,554 --> 00:05:51,303 So let's do that. So, the derivative is the limit as h 76 00:05:51,303 --> 00:05:58,699 approaches zero of, instead of sine theta plus h, it's sine theta times cosine h. 77 00:05:58,699 --> 00:06:05,849 This first term, plus cosine theta sine h minus sine theta from up here and this 78 00:06:05,849 --> 00:06:11,923 whole thing is divided by h. So, this is sine theta plus h minus sine 79 00:06:11,923 --> 00:06:16,271 theta all over h. Now, I can simplify this a bit. 80 00:06:16,271 --> 00:06:21,665 I've got a common factor of sine theta, so I can pull that out. 81 00:06:21,665 --> 00:06:29,129 This is the limit as h approaches zero, pull out that common factor of sine theta. 82 00:06:29,129 --> 00:06:36,553 What's left over is cosine h minus 1 over h plus, I've still got this term here, 83 00:06:36,553 --> 00:06:43,002 cosine theta sine h over h. I'll write that as cosine theta times sine 84 00:06:43,002 --> 00:06:44,684 h over h. All right. 85 00:06:44,684 --> 00:06:47,656 So this limit calculates the derivative of sine. 86 00:06:47,656 --> 00:06:52,398 Now, it's written as a limit of a sum, which is a sum of the limits, provided the 87 00:06:52,398 --> 00:06:55,617 limits exist. So, I can also note that sine theta and 88 00:06:55,617 --> 00:06:59,896 cosine theta are constants. Right, h is the thing that's wiggling, so 89 00:06:59,896 --> 00:07:03,117 I can pull those constants out of the limits as well. 90 00:07:03,117 --> 00:07:08,834 So what I'm left with is sine theta. Times the limit as h approaches 0 of 91 00:07:08,834 --> 00:07:14,741 cosine h minus 1 over h. Plus cosine theta times the limit as h 92 00:07:14,741 --> 00:07:21,297 approaches 0 of sine h over h. Now, how do I calculate these limits? 93 00:07:21,297 --> 00:07:29,123 Well, we've got to remember way back to when we were calculating limits a long, 94 00:07:29,123 --> 00:07:34,452 long time ago. We can calculate these limits by hand, 95 00:07:34,452 --> 00:07:41,032 using say, the squeeze theorem. And it happens that this first limit is 96 00:07:41,032 --> 00:07:45,000 equal to 0 and the second limit is equal to 1. 97 00:07:45,000 --> 00:07:51,378 So I'm left with 0 plus cosine theta. So, this is a limit argument, back from 98 00:07:51,378 --> 00:07:56,686 the definition of derivative that the derivative of sine is cosine. 99 00:07:56,686 --> 00:08:03,202 So, regardless of whether you think more geometrically or more algebraically, the 100 00:08:03,202 --> 00:08:08,922 derivative of sine is cosine and the derivative of cosine is minus sine. 101 00:08:08,923 --> 00:08:15,998 Now, once you believe this, there is some sort of weird things you might notice, 102 00:08:15,998 --> 00:08:21,222 like what if you differentiate sine a whole bunch of times? 103 00:08:21,222 --> 00:08:28,788 So the derivative of sine is cosine and the second derivative of sine, is the 104 00:08:28,788 --> 00:08:35,715 derivative of the derivative. It's the derivative of cosine which is 105 00:08:35,715 --> 00:08:41,031 minus sine. And the third derivative of sine is the 106 00:08:41,031 --> 00:08:49,416 derivative of the second derivative and the derivative of minus sine is minus 107 00:08:49,416 --> 00:08:53,891 cosine. And the fourth derivative of sine, well, 108 00:08:53,891 --> 00:09:00,106 that's the derivative of the third derivative, which is minus cosine. 109 00:09:00,106 --> 00:09:04,019 And the derivative of a cosine is minus sine. 110 00:09:04,019 --> 00:09:07,788 So, the derivative of minus cosine is sine. 111 00:09:07,788 --> 00:09:12,423 So the fourth derivative of sine is sine. So that's kind of interesting. 112 00:09:12,423 --> 00:09:16,369 If you differentiate sine four times, you get back to itself. 113 00:09:16,369 --> 00:09:21,457 And we already know a function that if you differentiate just once, it spits out 114 00:09:21,457 --> 00:09:24,559 itself again, e to the x is its own derivative. 115 00:09:24,560 --> 00:09:28,326 So as sort of a fun challenge, you might try to find a function f, so that if you 116 00:09:28,326 --> 00:09:31,748 differentiate it twice, you get back the original function, but if you 117 00:09:31,748 --> 00:09:35,312 differentiate it only once, you don't get back the original function. 118 00:09:35,312 --> 00:09:41,146 And if you can do this, you can ask the same question for even higher derivatives. 119 00:09:41,146 --> 00:09:46,016 We know a function whose fourth derivative is itself, but none of the earlier 120 00:09:46,016 --> 00:09:50,742 derivatives are the function again. Can you find a function whose third 121 00:09:50,742 --> 00:09:56,102 derivative is itself, but the first and second derivatives aren't the original 122 00:09:56,102 --> 00:09:59,715 function? That's a fun little game to play, anyway, 123 00:09:59,715 --> 00:10:03,998 it's a little challenge for you to try to find such a function.