TITLE: $\arctan$ of a square root as a rational multiple of $\pi$
QUESTION [10 upvotes]: I know that if $x$ is a rational multiple of $\pi$, then $\tan(x)$ is algebraic.
Is there a fairly simple way to express $x$ as $\pi\frac{m}{n}$, if $\tan(x)$ is given as a square root of a rational?

REPLY [3 votes]: The complete theory analogous to the one for rational tangents is worked out in

On Angles Whose Squared Trigonometric Functions are Rational
   by John H. Conway, Charles Radin, Lorenzo Sadun
http://arxiv.org/abs/math-ph/9812019