Question

The period of $f(x)=\cos \left(\frac{x}{3}\right)+\sin \left(\frac{x}{2}\right)$ is

• A. $2\pi$
• B. $4\pi$
• C. $8\pi$
• D. $12\pi$

Answer 1

Answer: (D)

Given, $f(x)=\cos \left(\frac{x}{3}\right)+\sin \left(\frac{x}{2}\right)$
Period of $\cos x$ and $\sin x$ are $2\pi$.
$\therefore$ Period of $f(x)=$ Period of
$\left[\cos \frac{x}{3}+\sin \frac{x}{2}\right]$
=Period of $\cos \frac{x}{3}+$ Period of $\sin \frac{x}{2}$
$=\frac{2\pi }{\left(\frac{1}{3}\right)}+\frac{2\pi }{\left(\frac{1}{2}\right)}=\frac{6\pi }{1}+\frac{4\pi }{1}$
$=\frac{\text{ LCM of }(6\pi ,4\pi )}{\text{ LCM of }(1,1)}=\frac{12\pi }{1}=12\pi$