I have a set of equations to solve which in the following form:
$ \cos(t_1 + t_2 + t_3 + t_4) + \sin(t_1 + t_2 - t_3 + t_4) + \cos(t_1 - t_4 + t_3 - t_5) + \sin(t_1 - t_2 + t_3 - t_5) + \cos(t_1 + t_3 + t_4 - t_5) + \sin(t_2 + t_3 - t_4 + t_5) = a $
$ \cos(t_1 + t_2 - t_3 - t_4) + \cos(t_1 + t_2 - t_4 + t_5) + \sin(t_1 + t_2 + t_4 + t_3) + \sin(t_1 - t_2 + t_4 - t_5) + \cos(t_2 + t_3 + t_4 + t_5) + \sin(t_1 - t_3 + t_4 - t_5) = b $
For $t_1, t_2, t_3, t_4, t_5$ scalar real variables and $a, b$ a real numbers. This example above is a specific (smaller) instance of the equations I need to solve, where each equation has dozens of cos/sin terms with linear combinations of 5 scalar variables as the argument. I need to be able to find all the possible solutions (or none) for such a form of problem.
Any recommendations on how I should approach such a problem? I am looking to use computational resources, and would be interested in a numerical/optimization approach of some type. Seeing that the problem is nonconvex, I'm at a bit of a loss.