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

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Solution

Simplification

Given: $sin x + sin 3 x + sin 5 x = 0$

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

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

$\Rightarrow 2 sin (\frac{6 x}{2}) cos (\frac{- 4 x}{2}) + sin 3 x = 0$

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

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

$sin 3 x = 0$ or $2 cos 2 x + 1 = 0$

$sin 3 x = 0$ or $cos 2 x = - \frac{1}{2}$

General solution for $sin 3 x = 0$

$sin 3 x = 0$

General solution is $3 x = n π, n \in Z$

$x = \frac{n π}{3}$ where $n \in Z$

General solution for $cos 2 x = - \frac{1}{2}$

$cos 2 x = - \frac{1}{2}$

$\Rightarrow cos 2 x = cos (\frac{2 π}{3})$

We know that general solution for $cos x = cos y$ is $x = 2 n π \pm y, n \in Z$

So, the general solution is $2 x = 2 n π \pm \frac{2 π}{3}, n \in Z$

$x = n π \pm \frac{π}{3}$ where $n \in Z$

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