Solve the equation and Find the General Solution.

Question

$$\sin 2x + \sin x = 0$$

This equation is getting on my nerves, partly because I think I have found the right answer and partly, my answer doesn't match in any one of the options given.

Answer 1

$$ \sin\left(2x\right)+\sin\left(x\right)=0 \Leftrightarrow \sin\left(x\right)\left(1+2\cos\left(x\right)\right)=0 $$ So the solutions are those of $\sin\left(x\right)=0$ so $x=k \pi$ and $\displaystyle \cos\left(x\right)=-\frac{1}{2}$ so also $\displaystyle x=\frac{2\pi}{3}+k\pi$ and $\displaystyle x=-\frac{2\pi}{3}+k\pi$