Zero-mean cosine polynomials which are non-negative for as long as possible

For a given integer n, all zero-mean cosine polynomials of order at most n which are non-negative on [0, (n/(n+l))pi] are found, and it is shown that this is the longest interval [0,theta] on which such cosine polynomials exi st. Also, the longest interval [0, theta] 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 [theta, pi] on which there are non-positive cosine p olynomials of degree n are solved. For both of these problems, all extremal polynomials are found. Application s of these polynomials to Diophantine approximation are suggested.