Solve: Sin 2x + sin 4x + sin 6x = 0. - Vidyadip

Question

Solve: Sin 2x + sin 4x + sin 6x = 0.

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Answer

We have,
$sin 2 x+sin 4 x+sin 6x = 0$
$\Rightarrow$ $sin 4 x+(sin 2 x+sin 6x) = 0$
$\Rightarrow$ $sin 4 x+2 sin 4 x\ cos 2x = 0$
$\Rightarrow$ $sin 4 x(1+2 cos 2x) = 0$
$\Rightarrow$ sin 4 x=0 or ,1+2 cos 2 x=0 $\Rightarrow$ sin 4x = 0 or $\cos 2 x=-\frac{1}{2}$
Now, $\sin 4 x=0 \Rightarrow 4 x=n \pi, n \in Z \Rightarrow x=\frac{n \pi}{4}, n \in Z$
And, $\cos 2 x=-\frac{1}{2}$
$\Rightarrow \quad \cos 2 x=\cos \frac{2 \pi}{3}$
$\Rightarrow \quad 2 x=2 m \pi \pm \frac{2 \pi}{3}, m \in Z$
$\Rightarrow \quad x=m \pi \pm \frac{\pi}{3}, m \in Z$
Hence, $x=\frac{n \pi}{4} or, x=m \pi \pm \frac{\pi}{3}$ where m, n $\in$Z.

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