Question

The period of the function $f\left(x\right)=\sin \left\{\sin \left(\frac{x}{5}\right)\right\}$ is:

• A. $2\mathrm{\pi }$
• B. $\frac{2\mathrm{\pi }}{5}$
• C. $10\mathrm{\pi }$
• D. $5\mathrm{\pi }$

Answer 1

The correct option is C

$10\mathrm{\pi }$

Explanation for the correct option:

Compute the period of the given function.

Given function, $f\left(x\right)=\sin \left\{\sin \left(\frac{x}{5}\right)\right\}$.

If $f\left(x\right)$ is a Periodic function with Period $T$ and $g\left(x\right)$ is any function such that range if $f$ is a proper subset of the domain if $g$, then $g\left(f\right(x\left)\right)$ is Periodic with Period $T$

We know that the period of $\sin \left(x\right)$ is $2\mathrm{\pi }$.

So the period of $\sin \left(\frac{x}{5}\right)=2\mathrm{\pi }\times 5=10\mathrm{\pi }$.
Since the quantity inside the sine function varies with a period of $10\mathrm{\pi }$.
So, that period of $f\left(x\right)=\sin \left\{\sin \left(\frac{x}{5}\right)\right\}=10\mathrm{\pi }$.
Hence, Option $\left(\mathrm{C}\right)$ is the correct answer.