# How do you solve sin(2theta)+sintheta=0 on the interval 0<=theta<2pi?

Apr 27, 2015

$\theta = 0 , \frac{2 \pi}{3} , \pi \mathmr{and} \frac{4 \pi}{3}$

#### Explanation:

Use $\sin \left(2 \theta\right) = 2 \sin \theta \cos \theta$

Then factor and solve.

$\sin \left(2 \theta\right) + \sin \theta = 0$

$2 \sin \theta \cos \theta + \sin \theta = 0$

$\sin \theta \left(2 \cos \theta + 1\right) = 0$

$\sin \theta = 0$$\textcolor{w h i t e}{\text{sssss}}$ or $\textcolor{w h i t e}{\text{sssss}}$ $2 \cos \theta + 1 = 0$ so

$\sin \theta = 0$$\textcolor{w h i t e}{\text{sssss}}$ or $\textcolor{w h i t e}{\text{sssss}}$ $\cos \theta = - \frac{1}{2}$

$\theta = 0 , \frac{2 \pi}{3} , \pi \mathmr{and} \frac{4 \pi}{3}$