Solve the following equation:
$sin x - sin 2 x + sin 5 x + sin 8 x = 0.$

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Solution

$sin x - sin 2 x + sin 5 x + s i n 8 x = 0$
$2 sin 3 x cos 2 x + 2 cos 5 x sin 3 x = 0$
$2 sin 3 x (cos 2 x + cos 5 x) = 0$
$sin 3 x cos \frac{7 x}{2} cos \frac{3 x}{2} = 0$
So, $sin 3 x = 0,$ $cos \frac{7 x}{2} = 0$ and $cos \frac{3 x}{2} = 0.$
$∴$ $x = \frac{n x}{3}, x = (2 n + 1) \frac{π}{7}, x = (2 n + 1) \frac{π}{3}$

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