1 00:00:00,012 --> 00:00:07,312 >> [music] How can anyone compute sine? I mean nowadays, people use computers or 2 00:00:07,312 --> 00:00:12,335 caluculators. But by what mysterious force are these 3 00:00:12,335 --> 00:00:18,039 computational devices able to perform these computations? 4 00:00:18,040 --> 00:00:22,155 How does the machine know what sine of one is equal to? 5 00:00:22,155 --> 00:00:27,027 So, that's a good question. How do we compute sine of 1 radian? 6 00:00:27,027 --> 00:00:30,226 One way is to use a little bit of calculus. 7 00:00:30,226 --> 00:00:35,551 So, what does calculus tell us? So let's let f of equals sine x, and the 8 00:00:35,551 --> 00:00:41,503 derivative of x is cosine x, the value of f at 0 is 0, sine 00, and the value of the 9 00:00:41,503 --> 00:00:44,825 derivative at 0 is 1, because cosine 0 is 1. 10 00:00:44,825 --> 00:00:48,945 Now, I can use the derivative to approximate the functions value. 11 00:00:48,945 --> 00:00:53,361 The functions value near zero is the functions value at zero, plus how much I 12 00:00:53,361 --> 00:00:57,701 would be the input by, times the ratio of output change to input change, the 13 00:00:57,701 --> 00:01:02,452 derivative at zero. Well given this, in our specific case the 14 00:01:02,452 --> 00:01:09,152 value of function 0 and derivative at 0 is 1, and that means the functions value at 15 00:01:09,152 --> 00:01:15,052 h, for h very small, is about h. That means that sine of a very small angle 16 00:01:15,052 --> 00:01:20,559 is very close to that same value. Let's actually use some numbers. 17 00:01:20,560 --> 00:01:26,371 So very concretely, sine of say 1 32nd is really close to 1 32nd. 18 00:01:26,371 --> 00:01:32,046 Now, in addition to this, we've also got a double angle formula. 19 00:01:32,046 --> 00:01:38,624 The double angle formula for sine says that sine of 2x is 2 times sine x times 20 00:01:38,624 --> 00:01:43,227 cosine x. You can derive this formula from the angle 21 00:01:43,227 --> 00:01:47,825 sum formula for sine. Sine of x plus x gives you this. 22 00:01:47,825 --> 00:01:51,660 Alright. Another way to rewrite that a double angle 23 00:01:51,660 --> 00:01:55,239 formula is to write it entirely in terms of sine. 24 00:01:55,239 --> 00:01:58,896 Alright. Cosine is the square root of 1 minus sine 25 00:01:58,896 --> 00:02:02,715 squared x. Well, as long as the cosine is positive, 26 00:02:02,715 --> 00:02:06,199 this is true. So, if I'm working with a value of x, 27 00:02:06,199 --> 00:02:11,511 which cos sins positive, then sin of 2 x is 2 times sin x times the square root of 28 00:02:11,511 --> 00:02:15,311 1 minus sin squared x. We can put these pieces together. 29 00:02:15,311 --> 00:02:21,056 Well, here's what we do. I know that sin of 1 32nd is very close to 30 00:02:21,056 --> 00:02:27,049 1 32nd because 1 32nd is close to 0. And, sin of a number close to zero is 31 00:02:27,049 --> 00:02:29,121 about that. That same number. 32 00:02:29,121 --> 00:02:33,653 Then I can use this double-angle formula entirely in terms of sign to get an 33 00:02:33,653 --> 00:02:38,398 approximate value for sign of 1 16th by using my approximate value for sign of 1 34 00:02:38,398 --> 00:02:41,285 32nd. Then, I can use this double-angle formula 35 00:02:41,285 --> 00:02:46,046 again to get an approximate value for sign of 1 18th, and again to get an approximate 36 00:02:46,046 --> 00:02:50,531 value for sign of 1 4th, and again to get an approximate value for sign of 1 half, 37 00:02:50,531 --> 00:02:53,801 and then again to get an approximate value of sign of 1. 38 00:02:53,801 --> 00:03:00,987 Which is really quite close to the actual value of sine of 1. 39 00:03:00,987 --> 00:03:06,726 So, what happened? Calculus told us that sine of a small 40 00:03:06,726 --> 00:03:15,052 number, is close to that small number. The double angle formula let us transport 41 00:03:15,052 --> 00:03:21,718 that bit of information around zero to points much further away. 42 00:03:21,718 --> 00:03:25,477 All the way to approximate sine of 1.