TITLE: Are there infinitely many rational outputs for sin(x) and cos(x)?
QUESTION [14 upvotes]: I know this may be a dumb question but I know that it is possible for $\sin(x)$ to take on rational values like $0$, $1$, and $\frac {1}{2}$ and so forth, but can it equal any other rational values? What about $\cos(x)$?

REPLY [8 votes]: You may be interested in Niven's theorem, which is that the only rational values of $\sin x$ when $x$ is a rational multiple of $\pi$ (i.e. a rational number of degrees) are $0$, $\pm\frac12$ and $\pm1$.  Of course $\sin x$ takes many other rational values but these do require $x/\pi$ to be irrational.