How can simultaneous sinusoidal equations be solved? I have come across a set of simultaneous equations which I can't figure out how to solve. I have 3 equations and only two unknowns, but they are angular quantities and feature in the equations as sinusoidal functions of the angular quantities.
The system of equations is:
$$ \begin{Bmatrix}\cos\psi\sin\theta+\cos\theta\sin\phi\sin\psi \\ \sin\psi\sin\theta-\cos\psi\cos\theta\sin\phi \\ \cos\phi\cos\theta \end{Bmatrix} = \begin{Bmatrix} x \\ y \\ z \end{Bmatrix} $$
Where: x, y, z and psi are known and phi and theta are unknown.
Is it possible to rearrange these equations to solve for theta and phi using the other terms?

The system is not always solvable. The two first equations factor as
$$\tag{1} \begin{pmatrix} \cos\psi & \sin\psi \\ \sin\psi & -\cos\psi \end{pmatrix}
\begin{pmatrix} \sin\theta \\ \cos\theta \sin\phi \end{pmatrix}
= \begin{pmatrix} x \\ y \end{pmatrix} $$
where the first factor is just a reflection in a line through the origin, so
$$ \sin^2\theta + \cos^2\theta\sin^2\phi = x^2 + y^2 $$
Squaring the third equation gives
$$ \cos^2\theta \cos^2\phi = z^2 $$
and adding these equations give
$$ x^2+y^2+z^2 = \sin^2\theta + \cos^2\theta(\sin^2\phi + \cos^2\phi) = 1 $$
so your system only has a solution when $(x,y,z)$ is a point on the unit sphere.
If your constants satisfy this, a natural approach would be to divide out the reflection matrix in $(1)$, producing
$$ \begin{cases} \sin \theta = x' \\ \cos\theta\sin\phi = y' \\ \cos\theta \cos\phi = z \end{cases} $$
for some $x'$ and $y'$. From here you get $\theta = \arcsin x'$ and and $\phi = \arctan \frac{y'}{z}$, both up to a choice of quadrants that have to be made in a matching way, possibly just by trial and error.
