Question

Find all the values of $\theta$ satisfying the equation : $\sin \theta +\sin 5\theta =\sin 3\theta$ such that $0\le \theta \le \pi$.

Answer 1

$\sin \theta +\sin 5\theta =\sin 3\theta \Rightarrow \sin \theta -\sin 3\theta +\sin 5\theta =0\Rightarrow 3\sin \theta -16{\sin }^3\theta +16{\sin }^5\theta =0\Rightarrow \sin \theta (2\sin \theta -1)(1+2\sin \theta )(4{\sin }^2\theta -3)=0$
$\Rightarrow \sin \theta =0$ or $\sin \theta =\frac{1}{2}$ or $\sin \theta =-\frac{1}{2}$ or $\sin \theta =\pm \frac{\sqrt{3}}{2}$
For $0\le \theta \le \pi$
We get $\theta =0,\frac{\pi }{6},\frac{\pi }{3},\frac{2\pi }{3},\frac{5\pi }{6},\pi$