Question

Find the number of values of $\theta$ satisfying the equation $\sin 3\theta =4\sin \theta .\sin 2\theta .\sin 4\theta$ in $0\le \theta \le 2\pi$

Answer 1

$4\sin \theta \sin 2\theta \sin 4\theta =\sin 3\theta \Rightarrow 4\sin \theta \sin (3\theta -\theta )\sin (3\theta +\theta )=\sin 3\theta \Rightarrow 4\left[\sin \theta \right(\sin {}^{2}3\theta -\sin {}^2\theta \left)\right]=3\sin \theta -4\sin {}^3\theta \Rightarrow 4\sin \theta \sin {}^{2}3\theta -4\sin {}^3\theta =3\sin \theta -4\sin {}^3\theta \Rightarrow 4\sin \theta \sin {}^{2}3\theta =3\sin \theta \Rightarrow \sin \theta (4\sin {}^{2}3\theta -3)=0\Rightarrow \sin \theta =0or4\sin {}^{2}3\theta -3=0\Rightarrow Now,\sin \theta =0\Rightarrow x=n{\pi }_{1}nϵz\sin {}^{2}3\theta =\frac{3}{4}\sin {}^{2}3\theta ={\left(\frac{\sqrt{3}}{2}\right)}^2\sin {}^{2}3\theta =\sin {}^2\frac{\pi }{3}3\theta =m\pi \pm \frac{\pi }{3},mϵzx=\frac{m\pi }{3}\pm \frac{\pi }{9},mϵz\therefore x=n\pi orx=\frac{m\pi }{3}\pm \frac{\pi }{9},m,nϵz$