TITLE: Is there a general formula for $\sin( {p \over q} \pi)$?
QUESTION [14 upvotes]: Virtually everyone knows the basic values of the unit circle, $\sin(\pi) = 0; \ \ \sin({\pi  \over 2}) = 1; \ \ \sin({\pi \over 3}) = {\sqrt{3} \over 2} \\$
And other values can be calculated through various identities, like $\sin({\pi \over 8}) =\frac{1}{2} \sqrt{2 - \sqrt{2}}\\$
Does there exist a general formula for $\sin({p\over q} \pi)$ for rational ${p \over q}$ as an algebraic number?

REPLY [3 votes]: There is not so much a formula, as there is an algorithm, due to Gauss. In his Disquisitiones Arithmeticae (sorry, I couldn't find a free English translation), he gives a number-theoretic view of the roots of unity (and thus, trigonometric functions). In particular, he outlines in one section a recursive method for determining explicit radical expressions of roots of unity.
Modern methods that improve on the asymptotic complexity of Gauss's algorithm exist, however. In this paper, A. Weber presents an improvement of Gauss's original algorithm that hinges on an efficient way to evaluate the "recursive step" in the original method. A Maple implementation of this improved method is presented in the paper.