It all boils down to building a certain matrix from the polynomial coefficients and computing its eigenvalues. John Boyd did a lot of work in this area, these are some relevant papers:
Boyd, John P. "A Fourier companion matrix (multiplication matrix) with real-valued elements: Finding the roots of a trigonometric polynomial by matrix eigensolving." Numerical Mathematics: Theory, Methods and Applications 6.4 (2013): 586-599.
Boyd, John P. "Computing the zeros of a Fourier series or a Chebyshev series or general orthogonal polynomial series with parity symmetries." Computers & Mathematics with Applications 54.3 (2007): 336-349.
Boyd, John P. "A comparison of companion matrix methods to find roots of a trigonometric polynomial." Journal of Computational Physics 246 (2013): 96-112.
Boyd, John P. "Finding the zeros of a univariate equation: proxy rootfinders, Chebyshev interpolation, and the companion matrix." SIAM review 55.2 (2013): 375-396.