For which integer $n$, $\sin\left(\frac{\pi}{n}\right)$ can be a rational?

Question

When I studying the trigonometric functions, I sow that most of the values of $\sin\left(\dfrac{\pi}{n}\right)$ and $\cos\left(\dfrac{\pi}{n}\right)$ where $n\in\mathbb{N}$ are irrational.
How can we determine all the $n\in\mathbb{N}$ such that $\sin\left(\dfrac{\pi}{n}\right)$ or $\cos\left(\dfrac{\pi}{n}\right)$ is a rational.
I don't see any approach for this problem?

Answer 1

Niven's theorem says that the only rational values of $\sin x$ when $x$ is a rational multiple of $\pi$ (that is, a rational number of degrees) are $0$, $\pm\frac12$ and $\pm1$.

A simple complete proof is given in a paper by Olmsted: Rational Values of Trigonometric Functions, Amer. Math. Monthly 52 (1945), no. 9, 507–508.