How to solve $\sin x \cos y - \sin u \cos v= a$?

Question

Please help me to solve this problem.

If $x\in[0,\pi]$, $y\in[0,2\pi]$, $u\in[0,\pi]$, and $a\in[-2,2]$ are given, how to find $v\in[0,2\pi]$?

Is this correct $v=\cos^{-1}(\frac{\sin x \cos y-a}{\sin u})$?

Only for certain values of $x,y,u,$ and $a$, I am getting a real $v$. Most of the time I get complex values.

Thanks