Chapter 10.2, Problem 31E

To determine

Tips to match $\sin x$ and $\cos x$ with their correct series.

Explanation of Solution

Given information:

It is assumed that trigonometric functions are well known.

  $\sin x$ is an odd function.

Whereas,

  $\cos x$ is an even function.

In case of $\sin x$ , when x is replaced by −x, we get $-\sin x$ .

Whereas,

In case of $\cos x$ , when x is replaced by −x, it remains $\cos x$ .

This is due to the odd powers of all the terms of $\sin x$ and even powers of all the terms of $\cos x$ in Maclaurin series.

Since $\sin 0=0$

This implies

No constant term exists in the Maclaurin series expansion of $\sin x$ .

Since $\cos 0=1$

This implies

Constant term in Maclaurin series expansion of $\cos x$ is 1.