Question

Fundamental Period of the function $f\left(\theta \right)=\frac{\sin \theta +\sin 2\theta }{\cos \theta +\cos 2\theta },\theta \in R-\left\{\right(2n+1)\pi ,2m\pi \pm \frac{\pi }{3},n,m\in Z\}$ is

• A. $\frac{2\pi }{6}$
• B. $\pi$
• C. $\frac{\pi }{2}$
• D. $\frac{2\pi }{3}$

Answer 1

The correct option is D $\frac{2\pi }{3}$

$f\left(\theta \right)=\frac{\sin \theta +\sin 2\theta }{\cos \theta +\cos 2\theta }=\frac{2\sin \left(\frac{3\theta }{2}\right)\cos \left(\frac{\theta }{2}\right)}{2\cos \left(\frac{3\theta }{2}\right)\cos \left(\frac{\theta }{2}\right)}=\tan \left(\frac{3\theta }{2}\right)$

We know that,
Period of $f\left(x\right)=\tan (ax+b)$ is $\frac{\pi }{|a|}$
So, period of $\tan \left(\frac{3\theta }{2}\right)=\frac{\pi }{3/2}=\frac{2\pi }{3}$