TITLE: arctan of a square root as a rational multiple of pi
QUESTION [7 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 $\pi\ m/n$, if $tan(x)$ is given as a square root of a rational?

REPLY [5 votes]: If $x \in \mathbb{Q}$ and $tan^2(x\pi) \in \mathbb{Q}$, then $tan(x\pi) \in \{0, \pm\sqrt{3}, \pm\frac{1}{\sqrt{3}}, \pm 1 \}$.
Chapter 11 of the A Concrete Approach to Abstract Algebra has a simple proof of this fact.
Also, a similar question has proofs that can be extended to this case.