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) $\sin \frac{1}{2}$ (b) $\sin 1$ (c) $\sin \frac{\pi}{6}$ (d) $\cos (-0.5)$ (e) $\cos 1$ (f) $\cos \frac{\pi}{4}$

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) $\sin \frac{1}{2}$
(b) $\sin 1$
(c) $\sin \frac{\pi}{6}$
(d) $\cos (-0.5)$
(e) $\cos 1$
(f) $\cos \frac{\pi}{4}$

Key Concepts

Taylor Series Expansion

Taylor series expansion represents a function as an infinite sum of terms calculated from its derivatives at a single point. This concept underpins the idea of approximating functions like sine and cosine by summing a finite number of terms from their series expansions.

Polynomial Approximation

Polynomial approximation involves using a finite number of terms from a Taylor series to estimate the value of a function. This approach is particularly useful for functions that have no simple algebraic form, enabling computations by substituting well-understood polynomial expressions.

Error Analysis in Approximations

Error analysis in the context of polynomial approximations addresses the difference between the true value of the function and the value given by a truncated series. It considers remainder terms and helps to understand that the approximation error can vary depending on the function input and the degree of the polynomial used.

Transcript

00:01 Okay, so in a polynomial approximation of sine and cosine.

00:06 So you can see that the polynomial approximation of sine x is written as an approximations from using squigglys.

00:15 X minus x to third over three factorial plus x to the fifth over five factorial.

00:23 Now, we're just using three terms of this, so this is going to definitely give us an error.

00:29 However, this is a mclaurin polynomial that goes infinitely.

00:37 So if we kept going, we'd have a minus x to the 7th over 7th, a factorial plus x to the 9th over 9 factorial.

00:43 And it can go infinitely, and it actually is equivalent to sine x.

00:49 Now, only using the first three terms means that i will be off in my values, but that's what we are going to compare.

00:57 So what's the best thing to do is if i'm going to have to find sign of one half, you might want to throw one half into your calculator and store it as like a variable, like store it as a.

01:12 And then go ahead and do a minus a to the third over three factorial is six, you know, plus a to the fifth power over five factorial, which is 120.

01:23 And then you can be using different a values as we go along.

01:30 So it ends up that the approximation is 0 .47947 .27.

01:37 Where if you actually put sign of one half in your calculator, you get something very, very close.

01:43 Notice it's only the last digit that is different.

01:47 And so that's a pretty good approximation.

01:49 It's a little bit of an overest of it.

01:51 Okay, now we're going to be putting 1 z.

01:54 Again, you can store one in as a and then recall the old equation that you use because we're using the same equation three times.

02:03 So we get the 0 .84 -16666, the actual value of sign using our calculator is 0 .84 -1470...