Question

Find the number of roots of the equation $\sin \theta +\sin 5\theta =\sin 3\theta$ in $[0,\pi ]$

Answer 1

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

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

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

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

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

$\Rightarrow \sin 3\theta =0,2\cos 2\theta =1$

$\Rightarrow \sin 3\theta =0,\cos 2\theta =\frac{1}{2}$

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

$\Rightarrow \theta =\frac{n\pi }{3},\theta =\frac{\pi }{6},\frac{5\pi }{6}$

$\Rightarrow \theta =0,\frac{\pi }{3},\frac{2\pi }{3},\pi ,\theta =\frac{\pi }{6},\frac{5\pi }{6}$

$\therefore \theta =\{0,\frac{\pi }{3},\frac{2\pi }{3},\pi ,\frac{\pi }{6},\frac{5\pi }{6}\}$

Hence number of solutions $=6$