We propose a set of simple and fast algorithms for evalu-ating and using trigonometric expressions in the form F = d k=0 f k cos k π n , f k ∈ Q, d < n for a fixed n ∈ Z>0: comput-ing the sign of such an expression, evaluating it numerically and computing its minimal polynomial in Q[x]. As critical byproducts, we propose simple and efficient algorithms for performing arithmetic operations (multiplication, division, gcd) on polynomials expressed in a Chebyshev basis (with the same bit-complexity than in the monomial basis) and for computing the minimal polynomial of 2 cos π n in O(n 2 0) bit operations with n0 < n is the odd squarefree part of n. Within such a framework, we can decide if F = 0 in O(d(τ +d)) bit operations , compute the sign of F in O(d 2 τ) bit operations and compute the minimal polynomial of F in O(n 3 τ) bit operations, where τ denotes the maximum bit-size of the f k 's.