How to solve $\sin^3 x=\sin x\,$?

Question

$\sin^3 x=\sin x$

I have absolutely no idea what to do. I've tried graphing, and I have a little better of an understanding, but I am at a loss.

Answer 1

Subtract $\sin x$ from both sides:

$$\sin^3x-\sin x = 0 \quad\Longrightarrow \quad\sin x(\sin^2x - 1) = 0$$

From here, follow the steps outlined in your previous post.

Answer 2

if $\sin (x)=0$ then the statement is true.

Assume $\sin (x) \neq 0$.

We want to find $x$ such that $\sin ^3x = \sin x$. Divide by $\sin x$ to get

$\sin^2 x=1$, which implies either $\sin (x)=1$ or $\sin (x)=-1$. for what values of $x$ does this happen?