The number of possible solutions of $\sin heta+\sin 4 heta+\sin 7 heta=0, heta \in(0, \pi)$ are

Question

The number of possible solutions of $\sin 	heta+\sin 4 	heta+\sin 7 	heta=0, 	heta \in(0, \pi)$ are

Aicharya · Accepted Answer

$\begin{aligned} & \sin 7 	heta+\sin 	heta+\sin 4 	heta=0 \ & \Rightarrow 2 \sin 4 	heta \cos 3 	heta+\sin 4 	heta=0 \ & \Rightarrow \sin 4 	heta(2 \cos 3 	heta+1)=0 \ & \Rightarrow \sin 4 	heta=0 \quad 	ext { or } \cos 3 	heta=\frac{-1}{2} \ & \Rightarrow \sin 4 	heta=0 \quad 	ext { or } \cos 3 	heta=\cos \left(\frac{2 \pi}{3}ight) \ & \Rightarrow 4 	heta=\mathrm{n} \pi \quad 	ext { or } 3 	heta=2 \mathrm{n} \pi \pm \frac{2 \pi}{3} \ & \Rightarrow 	heta=\frac{\mathrm{n} \pi}{4} \quad 	ext { or } \quad 	heta=\frac{2 \mathrm{n} \pi}{3} \pm \frac{2 \pi}{9} \ & \quad 	heta=\frac{2 \pi}{9}, \frac{\pi}{4}, \frac{4 \pi}{9}, \frac{\pi}{2}, \frac{3 \pi}{4}, \frac{8 \pi}{9} \ldots[\because 	heta \in(0, \pi)] \end{aligned}$ $	herefore \quad$ Number of solutions $=6$

The number of possible solutions of $\sin \theta+\sin 4 \theta+\sin 7 \theta=0, \theta \in(0, \pi)$ are

Solution