Chapter 0.6, Problem 59E

To determine

To show: the given function is periodic and find its period.

Answer to Problem 59E

Periodic with period $2\pi$

Explanation of Solution

Given information:

The given function is $y={\sin }^{3}x$ .

Consider the function:

  $\begin{array}{c}y={\sin }^{3}x\\ ={\left(\sin x\right)}^3\end{array}$

That is this function is obtained by raising $\sin x$ to 3 which is an odd power.

This gives that ${\sin }^{3}x$ will also range the same as $\sin x$ , that is from -1 to 1. And also both $\sin x$ and ${\sin }^{3}x$ will have same period, that is $2\pi$ .

This can also be observed by looking at the graph of the function $y={\sin }^{3}x$, which is given below:

  

Observing the graph it is clear that the function is periodic with period $2\pi$ .