Question

Find $a$ if the period of $f\left(x\right)=\tan 3x+\sin \left(\frac{x}{3}\right)$ is $3a\pi$.

Answer 1

Period for $\tan 3x$ is $\frac{\pi }{3}$
Period for $\sin \frac{x}{3}$ is $|2\pi \times \frac{3}{1}|=6\pi$
Thus, LCM of $\frac{\pi }{3}$ and $\frac{6\pi }{1}=\frac{6\pi }{1}$
Thus, $f\left(x\right)$ is periodic with period $6\pi$.