Question

The period of the function $f\left(\theta \right)=4+4{\sin }^3\theta -3\sin \theta$ is

• A. $\frac{2\pi }{3}$
• B. $\frac{\pi }{3}$
• C. $\frac{\pi }{2}$
• D. $\pi$

Answer 1

The correct option is A $\frac{2\pi }{3}$

Given, $f\left(\theta \right)=4+4{\sin }^3\theta -3\sin \theta$

$=4-(3\sin \theta -4{\sin }^3\theta )$

$=4-\sin 3\theta$

$\therefore f\left(\theta \right)=4-\sin 3\theta$

and $f(\theta +\frac{2\pi }{3})=4-\sin \left[3(\theta +\frac{2\pi }{3})\right]$

$=4-\sin (3\theta +2\pi )=4-\sin 3\theta$

Hence, $f\left(\theta \right)$ is periodic with period $\frac{2\pi }{3}$