Find the general solution of the equation $sin 2 x + sin 4 x + sin 6 x = 0$

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Solution

GIven $sin 2 x + sin 4 x + sin 6 x = 0$

$\Rightarrow (s i n 2 x + s i n 6 x) + s i n 4 x = 0$

$\Rightarrow 2 s i n 4 x c o s 2 x + s i n 4 x = 0$ $[∵ sin C + sin D = 2 sin \frac{C + D}{2} cos \frac{C - D}{2}]$

$\Rightarrow s i n 4 x (2 c o s 2 x + 1) = 0$

$\Rightarrow s i n 4 x = 0$

$\Rightarrow 4 x = n π$

$\Rightarrow x = \frac{n π}{4}, n ϵ Z$

$\Rightarrow (2 c o s 2 x + 1) = 0$

$\Rightarrow c o s 2 x = \frac{- 1}{2} = cos \frac{2 π}{3}$

$\Rightarrow c o s 2 x = cos (π - \frac{π}{3})$

$\Rightarrow 2 x = 2 m π + \frac{2 π}{3}; 2 m π - \frac{2 π}{3}, m ϵ Z$

$\Rightarrow x = m π + \frac{π}{3}; m π - \frac{π}{3}, m ϵ Z$

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