# Trigonometric Diophantine equation

Is there a general method to solve the equation $$P(x_1,x_2,...,x_n)=0$$ with $$P$$ is a polynomial in $$n$$ variables with integer coefficients and $$x_k=\cos(q_k\pi)$$ with $$q_k$$ is a rational number?

This type of equation appears in the problem of classifying all tetrahedra whose dihedral angles are multiples of $$\pi$$ and I want to know how to solve this kind of problem generally.

• If you know a lower bound for the denominator of all $q_k$, then it can be written as a diophantine equation in a cyclotomic field. There isn't a "general" method, but many methods that can help depending on the shape of the equation. Sep 27 at 8:43