It is correct, but not useful. For example, if you need to solve sin(x)=0.5, you find that x=pi/6 is a solution, and it's usually sufficient. Formally, the solution is x=pi/6+n2pi, n being any natural number (with negatives), just like you suggest for theta.
The goal is to find all solutions for x, so the correct answer must include them all. You do not need or want to find all possible values for theta. One value is enough, and you choose the one that is most readable.
Does that answer your question, or did I miss your point entirely?
(°v°)
