Question

Find the period of the following functions.$\frac{\sin \theta +\sin 2\theta }{\cos \theta +\cos 2\theta }$

Answer 1

$\frac{\sin \theta +\sin 2\theta }{\cos \theta +\cos 2\theta }=\frac{2\sin \frac{3\theta }{2}\cos \frac{\theta }{2}}{2\cos \frac{3\theta }{2}\cos \frac{\theta }{2}}$

$=\tan \frac{3\theta }{2}$

We know that $\tan x$ is periodic with fundamental period $\pi$.

So, $\tan \frac{3\theta }{2}$ is periodic with fundamental period T, such that

$\frac{3T}{2}=\pi$

$T=\frac{2\pi }{3}$