I have the following three equations:
$$\cos\theta \left(\cos\psi - k_3\sin\psi\right) = k_1$$
$$\sin\phi\sin\theta\cos\psi - \cos\phi\sin\psi - k_3\left(\cos\phi\cos\psi + \sin\phi\sin\theta\sin\psi\right) = k_2 $$
$$\frac{\sin\phi\sin\psi + \cos\phi\sin\theta\cos\psi}{\cos\phi\sin\theta\sin\psi - \sin\phi\cos\psi} = k_3$$
What is the best way (numerically or symbollically) to solve these equations for the unknown variables $\phi$, $\theta$ and $\psi$?
$k_1$, $k_2$ and $k_3$ are known parameters.
I have tried simple numerical approaches without great success.
let's say three equations are <1><2><3>, from <3>, $\sin\phi(\sin\psi +k_3 cos\psi)+\cos\phi\sin\theta(cos\psi-k_3sin\psi)=0 \to tan\phi(\sin\psi +k_3 cos\psi)+k_1tan\theta=0$
from<2>,$sin\phi sin\theta(cos\psi-k_3sin\psi)-cos\phi(sin\psi+k_3cos\psi)=k_2 \to sin\phi ( k_1tan\theta-ctn\phi(sin\psi+k_3cos\psi))=k_2 \to sin\phi k_1tan\theta(1+ctn^2\phi)=k_2 \to k_1tan\theta=k_2sin\phi \to sin\phi\left(\dfrac{\sin\psi +k_3 cos\psi}{cos\phi}+k_2\right)=0$
so we have $sin\phi=0$ or $\dfrac{\sin\psi +k_3 cos\psi}{cos\phi}+k_2=0$
now you can go further from here.