# Question #54096

Oct 9, 2015

$x = n \frac{\pi}{2} \textcolor{w h i t e}{\text{XXXXXXX}}$ or
$x = \frac{2 \pi}{3} + n 2 \pi \textcolor{w h i t e}{\text{XXX}}$ or
$x = \frac{4 \pi}{3} + n 2 \pi$

#### Explanation:

Since $\sin \left(3 x\right) + \sin \left(x\right)$
$= 2 \sin \left(2 x\right) \cos \left(x\right) \textcolor{w h i t e}{\text{XXX}}$(See note below if this is not obvious)

$\implies \sin \left(x\right) + \sin \left(2 x\right) + \sin \left(3 x\right)$
$\textcolor{w h i t e}{\text{XXX}} = \left[\sin \left(3 x\right) + \sin \left(x\right)\right] + \sin \left(2 x\right)$
$\textcolor{w h i t e}{\text{XXX}} = \left[2 \sin \left(2 x\right) \cos \left(x\right)\right] + \sin \left(2 x\right)$
$\textcolor{w h i t e}{\text{XXX}} = \sin \left(2 x\right) \left(2 \cos \left(x\right) + 1\right)$

$\implies$ the equation reduces to
$\sin \left(2 x\right) = 0$ or $\cos \left(x\right) = - \frac{1}{2}$

If $\sin \left(2 x\right) = 0$
$\textcolor{w h i t e}{\text{XXX}} \implies 2 x = n \pi$ .

If $\cos \left(x\right) = - \frac{1}{2}$
$\textcolor{w h i t e}{\text{XXX}}$(within the interval $\left[0 , {360}^{\circ}\right) = \left[0 , 2 \pi\right)$)
$\textcolor{w h i t e}{\text{XXX}} \implies x = {120}^{\circ}$ or $x = {240}^{\circ}$
$\textcolor{w h i t e}{\text{XXXXXX}} = \left(\frac{2 \pi}{3}\right)$ or $\left(\frac{4 \pi}{3}\right)$

Note
The transformation $\sin \left(3 x\right) + \sin \left(x\right) = 2 \sin \left(2 x\right) \cos \left(x\right)$
is based on the formula:
$\sin \left(A\right) + \sin \left(B\right) = 2 \sin \left(\frac{A + B}{2}\right) \cdot \cos \left(\frac{A - B}{2}\right)$