I can't help but wonder why you posted this. You understand, don't you, that most of the people who post or respond here are themselves mathematics majors if not professors? But you refer to trigonometry as "crap" and complain about professors who actually expect you toknowthe pre-requisite material. You are given five possible solutions and complain that it "takes ages to try all the options". Actually, it takes about 5 seconds to check each one so less than half a minute to check all.

In any case, one of the veryfirstthings you should have learned about sine and cosine is the where they are positive and where they are negative- and that's all you need to know to solve cos(x)= -sin(x). The precise values are irrelevant.