With known variables $(a,b,c) \in \mathbb{R}^3$, how can I find $(\theta, \psi, \varphi) \in \mathbb{R}^3$ with the following system?
$$ \left\{ \begin{aligned} a &= cos(\theta) \\ 2b &= sin(\psi-\theta)-sin(\psi+\theta) \\ 2c &= sin(\varphi+\theta)-sin(\varphi-\theta) \end{aligned} \right. $$
I've never learned to solve something like that. Also, the system solvers that I found on the Internet take too much time to compute the solution and give up.
Thanks to Don Thousand and Doug M. Here is the solution:
$$ \begin{aligned} &\left\{ \begin{aligned} a &= cos(\theta) \\ 2b &= sin(\psi-\theta)-sin(\psi+\theta) \\ 2c &= sin(\varphi+\theta)-sin(\varphi-\theta) \end{aligned} \right. \\[2ex] \Leftrightarrow\;&\left\{ \begin{aligned} \theta &= cos^{-1}(a) \\ b &= -cos(\psi)sin(\theta) \\ c &= cos(\varphi)sin(\theta) \end{aligned} \right. \\[2ex] \Leftrightarrow\;&\left\{ \begin{aligned} \theta &= cos^{-1}(a) \\ \psi &= cos^{-1}\biggl(-\frac{b}{sin(cos^{-1}(a))}\biggr) \\ \varphi &= cos^{-1}\biggl(\frac{c}{sin(cos^{-1}(a))}\biggr) \end{aligned} \right. \end{aligned} $$