# Question #6eeed

Jun 9, 2017

$x = \frac{k \pi}{2}$
$x = k \pi$

#### Explanation:

Transpose all terms to the left side:
sin 3x + sin x - 2sin 2x = 0 (1)
Use trig identity:
$\sin a + \sin b = 2 \sin \left(\frac{a + b}{2}\right) \cos \left(\frac{a - b}{2}\right)$
In this case:
sin 3x + sin x = 2sin (2x).cos x
The equation (1) becomes:
$2 \sin \left(2 x\right) . \cos x - 2 \sin \left(2 x\right) = 0$
$\left[2 \sin \left(2 x\right)\right] \left(\cos x - 1\right) = 0$
Either factor must be zero.
a. cos x = 1 --> x = 0, and $x = 2 \pi$
General answers: $x = 2 k \pi$
b. sin 2x = 0
2x = 0 --> $2 x = \pi$, --> $2 x = 2 \pi$ -->
x = 0, --> $x = \frac{\pi}{2}$, --> $x = \pi$.
General answers: $x = \frac{k \pi}{2}$