Chapter 13, Problem 1RE

To determine

Whether the function $f\left(x\right)=\cos x$ is periodic with a period of $\pi$ or not?

Answer to Problem 1RE

The statement is $\underset{\_}{\text{False}}$.

Explanation of Solution

It is given that the function $f\left(x\right)=\cos x$ is periodic with a period of $\pi$. 

For a function to be periodic, $f\left(x\right)=f\left(x+a\right)$ where a is the period of the function.

$\begin{array}{c}f\left(x+\pi \right)=\cos \left(x+\pi \right)\\ =-\cos x\left(\because \text{cos}\left(\pi +\theta \right)=\cos \theta \right)\\ =-f\left(x\right)\\ \ne f\left(x\right)\end{array}$

Thus, the function $f\left(x\right)=\cos x$ is not periodic with a period of $\pi$.

Therefore, the statement is $\underset{\_}{\text{False}}$.