$f\in c([-\pi,\pi])$ and if $\int_{-\pi}^\pi ~f(x) \sin (nx) \mathrm dx$ =$0$ then prove that $f(x) = 0$.

Question

$f\in c([-\pi,\pi])$ and if $\int_{-\pi}^\pi ~f(x) \sin(nx) \mathrm dx$ =$0$ then prove that $f(x) = 0$. Can anyone please help me to solve it?

I was trying to understand this $\int_{0}^{\pi}f(x)\cos nx =0$ for all non negative integer n then prove $f(x)$ is identically $0$ on $[0, \pi]$ where $cosnx$ is there in the place of $sinnx$. But I got some questions.

1) How $f$ can extended to a even function?

2) what will happen when $sinnx$ is there in the place of $cosnx$?