Solving $2\sin(x) + 3\sin(2x) = 0$

Question

Solve for x:

$$2\sin(x) + 3\sin(2x) = 0 $$

$$2\sin(x)(1 + 3\cos(x)) = 0$$

Stuck here. The solution mentions some arccos function, but I need a detailed explanation on this one.

Answer 1

If you have that $$2\sin{(x)}(1+3\cos{(x)})=0$$ then one or both of the factors must be equal to zero, hence either $$2\sin{(x)}=0$$ $$\sin{(x)}=0$$ $$x=\pi k $$ or $$1+3\cos{(x)}=0$$ $$\cos{(x)}=-\frac13$$ $$x=\pm\arccos{\left(-\frac13\right)}+2\pi k$$ where $k$ is an arbitrary integer. The function $\arccos{(x)}$ gives the value of $y\in\left[0,\pi\right]$ such that $\cos{(y)}=x$.

Answer 2

Hint: if the product of two numbers is $0$, at least one of them is $0$.