The technical term for what you want to do is root isolation or root bracketing. One way to approach this to find the minimal distance between the roots, like you are suggesting, and also a large enough bounded interval to contain all the roots. This idea was in fact used early on in the history of root isolation for real polynomials. However, these techniques have gotten more sophisticated with time. I imagine the situation would be similar for trigonometric polynomials.

Here's a reference that seems to discuss root isolation precisely for trigonometric polynomials:

Real zero isolation for trigonometric polynomials, by Achim Schweikard
ACM Transactions on Mathematical Software 18 350-359 (1992)
http://dx.doi.org/10.1016/0003-4916(78)90225-7

The technical term for what you want to do is root isolation or root bracketing. One way to approach this to find the minimal distance between the roots, like you are suggesting, and also a large enough bounded interval to contain all the roots. This idea was in fact used early on in the history of root isolation for real polynomials. However, these techniques have gotten more sophisticated with time. I imagine the situation would be similar for trigonometric polynomials.

Here's a reference that seems to discuss root isolation precisely for trigonometric polynomials:

Real zero isolation for trigonometric polynomials, by Achim Schweikard
ACM Transactions on Mathematical Software 18 350-359 (1992)
http://dx.doi.org/10.1145/131766.131775