How can you solve 5 equations that only consist of 3 undermined variables?

Suppose that there is a system of five trigonometric equations, among which are three indefinite parameters (eg, x, y, z). In principle, only the solutions of the three parameters that satisfy all five equations are permissible. I have tried to solve only three equations with "NSolve" and also with "FindRoot", of course the later command gives approximate solutions, NSolve took about 18 hours (the equations are too complicated in the expression) to solve the equation system, but the solutions do not satisfy the two remaining equations, indicating inconsistency. Can you please show me a way to solve the three equations, on the condition that the remaining two are fulfilled at the same time? That is, the code generates only the solutions that take all five equations into account. Any help in this regard will be of great help.

Thank you in advance.