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 <br/>
> ACM Transactions on Mathematical Software **18** 350-359 (1992) <br/>
> http://dx.doi.org/10.1145/131766.131775