Question

Find the general solution for the following equation:

sin x + sin 3x + sin 5x = 0

Answer 1

sin x + sin 3x + sin 5x = 0

$\Rightarrow$ (sin 5x + sin x) + sin 3x = 0

$\Rightarrow$ 2 sin $\left(\frac{5x+x}{2}\right)\text{cos}\left(\frac{5x-x}{2}\right)$ + sin 3x = 0

$\Rightarrow$ 2 sin 3x cos 2x+ sin 3x = 0

$\Rightarrow$ sin 3x (2 cos 2x+1) = 0

$\Rightarrow$ Either sin 3x = 0 or 2 cos 2x+1 = 0

$\Rightarrow$ 3x = $n\pi$ or cos 2x =- $\frac{1}{2}$ = cos $\frac{2\pi }{3}$, n $ϵ$ Z

$\Rightarrow$ x = $\frac{n\pi }{3}$ or 2x = $2n\pi \pm \frac{2\pi }{3}$, n $ϵ$ Z

$\Rightarrow$ x = $\frac{n\pi }{3}$ or x= $n\pi \pm \frac{\pi }{3}$ , n $ϵ$ Z.