You might be able to use a newton iteration for this problem. The MATLAB command is fmincon (you require the optimization toolbox). Before you have to reformulation the problem into a minimzation problem, which is quite easy, just write everything on the left side and apply a norm to this vector.
You might also try fsolve or fzero.
However, if there are multiple solutions (and it surely looks like that), consider restraining the solution space e.g. x,y,Z have to be within -pi to +pi. Then you have to use fmincon.
In case these methods do not work, you can choose one variable arbitraily and still obtain at least one solution.
Hope that helps, Ben