Niven's Theorem

Theorem
Consider the angles $\theta$ in the range $0 \le \theta \le \dfrac \pi 2$.

The only values of $\theta$ such that both $\dfrac \theta \pi$ and $\sin \theta$ are rational are:


 * $\theta = 0: \sin \theta = 0$


 * $\theta = \dfrac \pi 6: \sin \theta = \dfrac 1 2$


 * $\theta = \dfrac \pi 2: \sin \theta = 1$

Proof
It is suspected that this result is considerably older, and may date back as far as Charles Hermite.