Chapter 9.2, Problem 31E

To determine

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

Answer to Problem 31E

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

Constant term exists in the Maclaurin series expansion of $\cos x$ .

Explanation of Solution

Given information:

It is given that $\sin x$ is an odd function and $\cos x$ is an even function.

Calculation:

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

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

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 the Maclaurin series.

  $\sin x=0$

  $\cos 0=1$

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

Constant term exists in the Maclaurin series expansion of $\cos x$ .