Question

If $f\left(x\right)=\sin \frac{x}{3}+\cos \frac{3x}{10}$ and $f(n\pi +x)=f\left(x\right),$ then the least positive integral value of $n$ is

Answer 1

Since $f(n\pi +x)=f\left(x\right)$ which means $n\pi$ is the period of $f\left(x\right).$
Period of $\sin \frac{x}{3}$ is $6\pi$ and period of $\cos \frac{3x}{10}$ is $\frac{20\pi }{3}.$

Now, period of $\sin \frac{x}{3}+\cos \frac{3x}{10}$
$=\text{LCM of}(6\pi ,\frac{20\pi }{3})$
$=\frac{\text{LCM of}(6\pi ,20\pi )}{\text{HCF of}(1,3)}$
$=60\pi$
Since $f(n\pi +x)=f\left(x\right),$
$\therefore n=60$