Is there any known (symbolic) method that solves a system of equations/inequalities that have trigonometric functions on the left-hand side of the system? 


Ex) Find $x,y,\theta \in \mathbb{R}$ that satisfy


\begin{align}
2x + y + 3\cos(\theta) - 2\sin(\theta) \le& 0 \\\
x - y + 4\cos(\theta) + 2\sin(\theta) \le& 0
\end{align}



Solving a system of equations/inequalities that have trigonometric functions on the left-hand side
Using, e.g., the sin function, one can write a system of inequalities in a given variable $x$ that is satisfied if and only if $x$ is an integer. Therefore, an algorithm for solving inequalities of the kind you asked about would give an algorithm of finding all integer solutions to an arbitrary system of inequalities. This would contradict the negative solution to Hilbert's Tenth Problem. See <http://en.wikipedia.org/wiki/Hilbert%27s_tenth_problem>


