Question

Solve the equation $\sin \theta +\sin 5\theta =\sin 3\theta ,0\le \theta \le \pi .$

Answer 1

$\sin \theta +\sin 5\theta =\sin 3\theta ,0\le \theta \le \pi$

$\sin \theta +\sin 5\theta =\sin 3\theta$

$\Rightarrow 2\sin \left(\frac{\theta +5\theta }{2}\right)\cos \left(\frac{\theta -5\theta }{2}\right)=\sin 3\theta$

$\Rightarrow 2\sin 3\theta \cos 2\theta =\sin 3\theta$

$\Rightarrow \sin 3\theta (2\cos 2\theta -1)=0$

$\Rightarrow \sin 3\theta =0$ or $2\cos 2\theta -1=0$

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

$\Rightarrow \theta =\frac{n\pi }{3}$ $\Rightarrow 2\theta =\frac{\pi }{3}+2n\pi ,2\theta =\frac{5\pi }{3}+2\pi n$

$\Rightarrow \theta =\frac{\pi }{6}+n\pi ,\theta =\frac{5\pi }{6}+n\pi$

$\because 0\le \theta \le \pi$ the values of $\theta =0,\underset{}{\frac{\pi }{3},\frac{2\pi }{3},\pi },\underset{}{\frac{\pi }{6},\frac{5\pi }{6}}$

$\sin 3\theta =0$ $2\cos \theta -1=0$