Number of solutions of the equation
$| sin x | = sin x - 2 cos x$ in $[0, 4 π]$ is

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Solution

Case 1: $sin x \geq 0$
$sin x = sin x - 2 cos x \Rightarrow cos x = 0$
$\Rightarrow x = \frac{π}{2}, \frac{3 π}{2}, \frac{5 π}{2}, \frac{7 π}{2}$
Since, $sin x \geq 0 \Rightarrow x = \frac{π}{2}, \frac{5 π}{2}$

Case 2: $sin x < 0$
$- sin x = sin x - 2 cos x$
$\Rightarrow - 2 sin x = - 2 cos x \Rightarrow tan x = 1$
$\Rightarrow x = \frac{π}{4}, \frac{5 π}{4}, \frac{9 π}{4}, \frac{13 π}{4}$
Since, $sin x < 0 \Rightarrow x = \frac{5 π}{4}, \frac{13 π}{4}$
$∴$ Number of solutions is $4$ .

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