Principal general solutions in the complex plane found as follows:
Substitute sin() & cos() functions with their equivalents in complex plane:
to obtain:
Solve to obtain the 4 principal roots:
Similarly, the 2nd equation will become:
with 6 principal roots:
I'll try to explain somewhat what is going on. On the complex plane we have that
If you let we get this representation
Next, we substitute these in your equations. The first one is
After we multiply everything by and expand, the equation becomes
The second equation becomes
After we multiply everything by and expand, the equation becomes
Solving each of these equations gives you the roots that MaxJasper wrote (I used a math program to find them for me). These equations need to hold simultaneously, so the only roots you accept are ones that solve both equations. If you inspect them, you will see that only is common to both. Now, we have that , so you are looking for solutions of this new equation
Knowing a bit about the complex exponential, you will immediately see that the solutions to this new equation are
As a final step you need check which of these solves the original equations, if any. This is indeed the case.