For a given integer n, all zero-mean cosine polynomials of order at most n which are non-negative on [0,(n/(n+1))π] are found, and it is shown that this is the longest interval [0,θ] on which such cosine polynomials exist. Also, the longest interval [0,θ] on which there is a non-negative zero-mean cosine polynomial with non-negative coefficients is found.

As an immediate consequence of these results, the corresponding problems of the longest intervals [θ,π] on which there are non-positive cosine polynomials of degree n are solved.

For both of these problems, all extremal polynomials are found. Applications of these polynomials to Diophantine approximation are suggested.