The Fundamental Theorem of Algebra states that "Any polynomial of degree $n$ ... has $n$ roots." Is there anything analogous for trigonometric equations? I've been solving some trigonometric equations, and solving some of the slightly more complex ones involves, what seems like guess and check, applying certain trig identities (namely $sin (\theta + 2\pi)$ or $sin(\pi - \theta) = sin (\theta)$) over and over again to check to see whether the resultants are within the given domain.
So, my question is, if we cant determine the exact solutions from a method other than guess and check, can we at least determine the number of solutions of a trigonometric equation given its range using a more mathematical procedure (i.e. no guess and check whatsoever)?