Question

The possible value of $\Theta ϵ(0,\pi )suchthatsin\left(\Theta \right)+sin\left(4\Theta \right)+sin\left(7\Theta \right)=0are:$

Answer 1

$\sin \theta +\sin 4\theta +\sin 7\theta =0$

$(\sin \theta +\sin 7\theta )+\sin 4\theta =0$

$2\sin \left(\frac{\theta +7\theta }{2}\right)\cos \left(\frac{\theta -7\theta }{2}\right)+\sin 4\theta =0$

$2\sin 4\theta \cos 3\theta +\sin 4\theta =0$

$\sin 4\theta (2\cos 3\theta +1)=0$

$\Rightarrow \sin 4\theta =0$ or $\cos 3\theta =\frac{-1}{2}$

$\Rightarrow 4\theta =n\pi$ or $3\theta =2n\pi \pm \frac{2\pi }{3}$

$\Rightarrow \theta =\frac{n\pi }{4}$ or $\theta =\frac{2n\pi }{3}+\frac{2\pi }{9}$

As $\theta \in (0,\pi )$

$\theta =\frac{\pi }{4}$ or $\frac{2\pi }{4}$ or $\frac{3\pi }{4}$

$\theta =\frac{2\pi }{9},\frac{2\pi }{3}-\frac{2\pi }{9},\frac{2\pi }{3}+\frac{2\pi }{9}$

Total number of solutions$=3+3=6$.