The solutions of $\sin x+\sin 5 x=\sin 3 x$ in $\left(0, \frac{\pi}{2}\right)$ are

The solutions of $\sin x+\sin 5 x=\sin 3 x$ in $\left(0, \frac{\pi}{2}\right)$ are
  1. $\frac{\pi}{4}, \frac{\pi}{10}$
  2. $\frac{\pi}{6}, \frac{\pi}{3}$
  3. $\frac{\pi}{4}, \frac{\pi}{12}$
  4. $\frac{\pi}{8}, \frac{\pi}{16}$

Solution

$\begin{aligned} & \sin x+\sin 5 x=\sin 3 x \\ & \Rightarrow 2 \sin 3 x \cos 2 x=\sin 3 x \\ & \Rightarrow \sin 3 x(2 \cos 2 x-1)=0 \\ & \Rightarrow \sin 3 x=0 \text { or } \cos 2 x=\frac{1}{2}=\cos \frac{\pi}{3} \\ & \Rightarrow 3 x=\mathrm{n} \pi \text { or } 2 x=2 \mathrm{n} \pi \pm \frac{\pi}{3} \\ & \Rightarrow x=\frac{\mathrm{n} \pi}{3} \text { or } x=\mathrm{n} \pi \pm \frac{\pi}{6}\end{aligned}$ $\Rightarrow x=\frac{\pi}{3}, \frac{\pi}{6} \quad \quad \ldots\left[\because x \in\left(0, \frac{\pi}{2}\right)\right]$

Asked in: MHT CET 2023 (13 May Shift 2)

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