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Sine values being rational
I'm guessing that if I look in Ivan Niven's elementary book on irrational numbers, I'll find the answer to this quickly, but I'm posting it here in case people find it useful.
For what rational values of $x/\pi$ is $\sin x$ rational?
Obviously $\sin 0$, $\sin (\pi/6)$, $\sin (\pi/2)$ and their counterparts in the other quadrants will do it. I believe I've seen it asserted by someone who should know, that those are the only ones. How is that proved?
Those are the only ones. I'll answer the question for cosine instead: $2 \cos \frac{2 \pi k}{n}$ is the sum of two algebraic integers $\zeta_n + \zeta_n^{-1} = e^{ \frac{2 \pi i k}{n} } + e^{ - \frac{2 \pi i k}{n} }$, hence an algebraic integer, so it is rational if and only if it is an integer. Hence $\cos \frac{2 \pi k}{n} = 0, \pm \frac{1}{2}, \pm 1$.
In fact, more can be said. $\mathbb{Q}(\cos \frac{2 \pi k}{n})$ is the real subfield of $\mathbb{Q}(\zeta_n)$, hence has degree $\frac{\varphi(n)}{2}$ over $\mathbb{Q}$.