I'm a high school calculus/precalculus teacher, so forgive me if the question is a little basic. One of my (very gifted) students recently came up with a construction yielding a quartic, one of whose roots was sin(80º) -- which led me to the startling discovery that this (and, indeed, all rational values of sine/cosine (in degrees; that is, rational multiples of π)) are algebraic.

I've come across a number of proofs that the numbers are algebraic since, which, as I understand it, goes back to complex roots of unity. What I -haven't- seen, and would very much like to see/understand, is some general method for generating/constructing polynomials (w/ integer coefficients) whose roots are sine/cosine of rational values (in degrees). (My student's method only works for 80º/10º, 70º/20º, and 75º/15º, unfortunately). Would much appreciate...