# Rational multiples of $\pi/2$ whose sines are also rational

Let $f(x)=\sin(x\frac{\pi}{2})$. Let $R$ the set of $x$ such that :

• $0\le x\le 1$
• $x \in \mathbb Q$
• $f(x) \in \mathbb Q$

Hence, $0\in R$ as $f(0)=0$. $1\in R$ as f(1)=1. And $\frac{1}{3}\in R$ as $f(\frac{1}{3})=\frac{1}{2}$.

Are they any other elements of $R$ ? How to find them all ?

Thanks !

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+1 Very Interesting question. –  Adam Rubinson Jan 10 '13 at 13:56
–  Hans Lundmark Jan 10 '13 at 15:25
Is there a proof of Niven's theorem? I can't seem to find it on any of those sites or links... –  Adam Rubinson Jan 10 '13 at 15:58
Actually there is a Proof Sketch in the "Talk" section of the wikipedia article. Thanks –  Adam Rubinson Jan 10 '13 at 16:28