# How do you find the solutions to 8sin^2(3x) = -7sin(3x)?

Oct 22, 2015

Solve 8sin^2 (3x) = - 7sin (3x)

Ans: $- {20}^{\circ} 35 \mathmr{and} {80}^{\circ} 35$

#### Explanation:

$8 {\sin}^{2} \left(3 x\right) + 7 \sin \left(3 x\right) = 0$
Put sin (3x) in common factor:
$\sin \left(3 x\right) \left(8 \sin 3 x + 7\right) = 0$

a. $\sin 3 x = 0$ --> 3x = 0 --> x = 0
$\sin 3 x = \pi$ --> $x = \frac{\pi}{3}$
$\sin 3 x = 2 \pi$ --> $x = \frac{2 \pi}{3}$
b. 8sin (3x) + 7 = 0 --> $\sin 3 x = - \frac{7}{8}$
3x = - 61.04 --> x = $- {20}^{\circ} 35$
3x = 180 - (-61.04) = 241.04 --> $x = {80}^{\circ} 35$

Check by calculator
x = 80.35 3x = 241.04 --> sin 3x = -0.875 --> ${\sin}^{2} \left(3 x\right) = 0.77$
$8 {\sin}^{2} \left(3 x\right) = 0.77 \left(8\right) = 6.12$ --> $- 7 \sin 3 x = 7 \left(0.875\right) = 6.125$. OK