For which $q\in\mathbb Q$ is $\sin(\frac\pi2q)$ rational?

Question

Do there exist rational numbers $q \in (0,1) \cap \mathbb Q$ such that $$\sin\left(\frac{\pi}{2}q\right) \in \mathbb Q\;?$$

Clearly if $q \in \mathbb Z$, yes. But what about the case $0 < q < 1$?

As $\sin(\pi/6) = 1/2$ we have $q = 1/3$ is a solution. Are there any others?

Answer 1

The only rationals $r$ such that $\sin(\pi r)$ is rational are those for which $ \sin(\pi r)$ is in $\{-1,-1/2,0,1/2,1\}$. This is because $2 \sin(\pi r)$ is an algebraic integer, and algebraic integers that are rational are ordinary integers.

Answer 2

$$q=\frac{1}{3}\Rightarrow \sin \left( \frac{\pi}{2} \frac{1}{3}\right) =\frac{1}{2}$$