Use the polynomial approximations of the sine and cosine...

Use the polynomial approximations of the sine and cosine functions in Exercise 103 to approximate the following function values. Compare the results with those given by a calculator. Is the error in the approximation the same in each case? Explain. (a) $sin\ \dfrac{1}{2}$ (b) $sin\ 1$ (c) $sin\ \dfrac{\pi}{6}$ (d) $cos(-0.5)$ (e) $cos\ 1$ (d) $cos\ \dfrac{\pi}{4}$

Use the polynomial approximations of the sine and cosine functions in Exercise 103 to approximate the following function values. Compare the results with those given by a calculator. Is the error in the approximation the same in each case? Explain.

(a) $sin\ \dfrac{1}{2}$

(b) $sin\ 1$

(c) $sin\ \dfrac{\pi}{6}$

(d) $cos(-0.5)$

(e) $cos\ 1$

(d) $cos\ \dfrac{\pi}{4}$

Transcript

00:02 Okay, for this problem we're going to use the approximation method given in the previous problem number 103, where we were told that we could approximate sign of x with x minus x cubed over 3 factorial plus x to the 5th over 5 factorial.

00:17 So in part a, let's suppose that x is 1 1⁄2.

00:20 So if we were just to calculate the sign of 1 half, we would get approximately 0 .47943.

00:29 And then if we're going to use the approximation formula, we would have one half minus one half cubed over three factorial plus one half to the fifth over five factorial.

00:47 And that gives us approximately 0 .47943 as well.

00:57 So at least to that many decimal places, they are very, very close.

01:00 I'm not going to assume that they are exactly the same.

01:03 We would have to look further at more decimal places.

01:06 Let's do the sign of 1.

01:09 So if you let x equal 1, the sign of 1, according to the calculator, is 0 .84147.

01:19 And then if you use the approximation formula, you have 1 minus 1 cubed over 3 factorial, plus 1 to the 5th over 5 factorial.

01:28 And that gives you approximately 0 .8467.

01:36 So you can see a little bit of a difference between the numbers in the what, 10 ,000s place, 10 ,000th place, and the 100 ,000th place.

01:47 And then let's take a look for part c at the sign of pi over 6.

01:52 So if we just put sign of pi over 6 into the calculator, we can get an exact value for that.

01:58 I do believe that gives us 0 .5.

02:02 So we can just make that exactly 0 .5, not even approximate.

02:07 Now if we approximate using our formula, we would have pi over 6 minus pi over 6 cubed over 3 factorial plus pi over 6 to the 5th over 5 factorial.

02:22 And the approximate value of that is 0 .5 .00, 021, etc.

02:34 So you can see that those are very, very close.

02:36 Not exactly the same, but very, very close...

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