Question

Fundamental period of the function $f\left(\theta \right)=|2\sin 3\theta +4\cos 3\theta |$ is

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

Answer 1

The correct option is D $\frac{\pi }{3}$

Period of $2\sin 3\theta$ is $\frac{2\pi }{3}$ and period of $4\cos 3\theta$ is $\frac{2\pi }{3}$.
Hence, the period of $2\sin 3\theta +4\cos 3\theta$ is the L.C.M. of period of $2\sin 3\theta$ and $4\cos 3\theta .$
L.C.M of $(\frac{2\pi }{3},\frac{2\pi }{3})=\frac{2\pi }{3}$
Since modulus is present, we should be checking for a smaller period as well.
Checking for $\frac{\pi }{3},$
$f\left(\theta \right)=|2\sin 3\theta +4\cos 3\theta |$
$\Rightarrow f(\frac{\pi }{3}+\theta )=|2\sin (\pi +3\theta )+4\cos (\pi +3\theta )|$
$\Rightarrow f(\frac{\pi }{3}+\theta )=|-2\sin 3\theta -4\cos 3\theta |=|2\sin 3\theta +4\cos 3\theta |$
$\Rightarrow f(\frac{\pi }{3}+\theta )=f\left(\theta \right)$
$\therefore$ Fundamental period of $|2\sin 3\theta +4\cos 3\theta |$ is $\frac{\pi }{3}$