• # question_answer The possible values of $\theta \in (0,\pi )$such that $\sin (\theta )+sin(4\theta )+sin(7\theta )=0$are :     AIEEE  Solved  Paper (Held On 11 May  2011) A) $\frac{\pi }{4},\frac{5\pi }{12},\frac{\pi }{2},\frac{2\pi }{3},\frac{3\pi }{4},\frac{8\pi }{9}$ B) $\frac{2\pi }{9},\frac{\pi }{4},\frac{\pi }{2},\frac{2\pi }{3},\frac{3\pi }{4},\frac{35\pi }{36}$ C) $\frac{2\pi }{9},\frac{\pi }{4},\frac{\pi }{2},\frac{2\pi }{3},\frac{3\pi }{4},\frac{8\pi }{9}$ D) $\frac{2\pi }{9},\frac{\pi }{4},\frac{4\pi }{9},\frac{\pi }{2},\frac{3\pi }{4},\frac{8\pi }{9}$

$\sin 4\theta +2\sin 4\theta \cos 3\theta =0$$\because$$\theta ,\in (0,\pi )$ $\sin 4\theta (1+2cos3\theta )=0$ $\sin 4\theta =0$                           or            $\cos 3\theta =-\frac{1}{2}$ $4\theta =n\pi ;n\in I$                or            $3\theta =2n\pi \pm \frac{2\pi }{3},n\in I$ $\theta =\frac{\pi }{4},\frac{\pi }{2},\frac{3\pi }{4}$                       or            $\theta =\frac{2\pi }{9},\frac{8\pi }{9},\frac{4\pi }{9}$