Question

If $\sin \theta +\sin 2\theta +sin3\theta +\sin 4\theta =0,$ then $\theta =$

• A.
• B.
• C.
• D.

Answer 1

The correct option is C

$\sin \theta +\sin 3\theta +\sin 2\theta +\sin 4\theta =0\Rightarrow 2\sin 2\theta .\cos \theta +2\sin 3\theta .\cos \theta =0$
$\Rightarrow 2\cos \theta (\sin 2\theta +\sin 3\theta )=0\Rightarrow 4\cos \theta .\sin \frac{5\theta }{2}.\cos \frac{\theta }{2}=0\Rightarrow \cos \theta =0or\cos \frac{\theta }{2}=0or\sin \frac{5\theta }{2}=0$
$\Rightarrow \theta =(2n+1)\frac{\pi }{2},\frac{\theta }{2}=(2n+1)\frac{\pi }{2},\frac{5\theta }{2}=n\pi \Rightarrow \theta =(2n+1)\frac{\pi }{2},or(2n+1)\pi or\frac{2n\pi }{5}.$