TITLE: When is $\sin(x)$ rational?
QUESTION [27 upvotes]: Obviously, there are some points (e.g. $\pi$, $30^\circ$) but I am unsure if there are more. How can it be proved that there are no more points, or, if there are, what those points will be?
EDIT: I largely meant to ask what is supposed in the first comment. Put another way, are there numbers such that $\sin x$ and $\arcsin x$ are both rational?

REPLY [12 votes]: There are infinitely many points where both the sine and the cosine are rational, namely
$$
\left( \frac{n^2-m^2}{n^2+m^2}, \frac{2mn}{n^2+m^2} \right).
$$
You can see that if $(x,y)=\text{that pair}$ then $x^2+y^2=1$, so it's on the unit circle.
Maybe a more interesting question is when do you have an angle that is a rational number of degrees (or, equivalently, a rational multiple of $\pi$ radians) for which the sine is rational.  I think in that case you get only the obvious ones: $\sin 0^\circ=0$, $\sin 30^\circ=1/2$, $\sin 90^\circ=1$, and the counterparts in the other quadrants.  I'm not sure right now how to prove that.