Analytic equations to make algorithm of 3D printer
As shown, the mechanics are under-constrained. You can't solve for theta because you have three degrees of freedom (X, Y, theta) and only two constraints (L1, L2). Gravity will tend to bias theta in a particular orientation, but the geometrical stiffness of this arrangement will be so low that it will not be possible to do 3D printing. 

To calculate the free-hanging orientation of theta, you will need to know the center of gravity of the end-effector, and solve a system of equations to find the angles and tensions for each cable that produce force vectors which sum to equilibrium with the gravity force vector through the COG. Unfortunately, the tensions will be a function of the angles, so it's not trivial to solve. As a hint, the virtual intersection of the two cables will be coincident with or directly above the COG in all equilibrium positions, and the horizontal components of the tensions in the two cables will be equal.