Find the general solution of $sin x + sin 3 x + sin 5 x = 0$

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Solution

We have, $sin x + sin 3 x + sin 5 x = 0$
$∴ (sin x + sin 5 x) + sin 3 x = 0$
$∴ 2 sin (\frac{x + 5 x}{2}) cos (\frac{5 x - x}{2}) + sin 3 x = 0$
$∴ 2 sin 3 x cos 2 x + sin 3 x = 0$
$∴ sin 3 x (2 cos 2 x + 1) = 0$
Either $sin 3 x = 0$ or $2 cos 2 x + 1 = 0$
i.e. $sin 3 x = 0$ or $cos 2 x = - \frac{1}{2}$

Now, $cos 2 x = - cos \frac{π}{3}$
$∴ cos 2 x = cos (π - \frac{π}{3})$
$∴ cos 2 x = cos \frac{2 π}{3}$

$∴ sin 3 x = 0$ or $cos 2 x = cos \frac{2 π}{3}$
$3 x = n π, n \in Z$ or $2 x = 2 m π \pm \frac{2 π}{3}$ where $m \in Z$
Hence, $x = \frac{n π}{3}$ or $x = m π \pm \frac{π}{3}$ , where $n, m \in Z$ .

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