If the general solution set of $\sin x+3 \sin 3 x+\sin 5 x=0$ is S , then $\{\sin \alpha / \alpha \in S\}=$

If the general solution set of $\sin x+3 \sin 3 x+\sin 5 x=0$ is S , then $\{\sin \alpha / \alpha \in S\}=$
  1. $\{1,-1,0\}$
  2. $\left\{\frac{1}{2}, \frac{-1}{2}, 0,1,-1\right\}$
  3. $\left\{\frac{\sqrt{3}}{2}, 0, \frac{-\sqrt{3}}{2}\right\}$
  4. $\left\{1,-1, \frac{\sqrt{3}}{2}, 0, \frac{-\sqrt{3}}{2}\right\}$

Solution

$\begin{aligned} & \text { } \sin x+\sin 5 x+3 \sin 3 x=0 \\ & \Rightarrow 2 \sin 3 x \cos 2 x+3 \sin 3 x=0 \\ & \Rightarrow \sin 3 x(2 \cos 2 x+3)=0 \\ & \Rightarrow \sin 3 x=0\left(\because \cos 2 x \neq \frac{-3}{2}\right) \\ & \Rightarrow 3 x=0, \pi, 2 \pi, 3 \pi, 4 \pi, 5 \pi \\ & \Rightarrow x=0, \frac{\pi}{3}, \frac{2 \pi}{3}, \pi, \frac{4 \pi}{3}, \frac{5 \pi}{3} \Rightarrow \sin \alpha \in\left\{0, \frac{\sqrt{3}}{2}, \frac{-\sqrt{3}}{2}\right\}\end{aligned}$

Asked in: AP EAMCET 2024 (21 May Shift 1)

Practice more Functions questions on Aicharya