Issue:
I am trying to solve the following system of nonlinear equations for the unknown variables: $x$, $z$ and $\beta$. The remaining variables are known values.
$$a=u(s^2+(x\cos\beta\ )^2+(z\sin\beta\ )^2\ ) \tag{eq. 1}\label{eq1}$$ $$b=u(y^2+(x\sin\beta\ )^2+(z\cos\beta\ )^2\ ) \tag{eq. 2}\label{eq2}$$ $$c=y \cdot s+(x^2-z^2\ )\ \ 1/2\ \sin2\beta \tag{eq. 3}\label{eq3}$$
It can be assumed that all variables are real numbers and the following may also be assumed positive: $a$, $b$ and $u$.
Any guidance on how to best approach this question would be highly appreciated.
What I have tried so far:
I have tried solving it with substitution, simplifying wherever I can, but the equations expand drastically and become too complex to handle for me. An alternate route I tried was using a symbolic solver in python (sympy), but this also struggles with coming up with a solution.
Moreover, I have tried adding eq. 1 & 2 together, which produces the following more "simple" equation utilizing Pythagorean identity, but issues with complex expansion still persists:
$$a+b=u\left(y^2\ +s^2+x^2+z^2\ \right) \tag{eq. 4}\label{eq4}$$
From the third equation:
$$ z^2 = x^2 + \frac{s\,y-c}{\sin\beta\cos\beta} $$
which replaced in the first and second equation leads to:
$$ a = u\left(s^2+x^2+(s\,y-c)\tan\beta\right), \quad \quad b = u\left(x^2+y^2+(s\,y-c)/\tan\beta\right). $$
Therefore, by subtracting the latter two equations, we get:
$$ a-b=u\left(s^2-y^2+(s\,y-c)\tan\beta-(s\,y-c)/\tan\beta\right) $$
that is, solving with respect to $\tan\beta$:
$$ \tan\beta = \frac{\left(\frac{a-b}{u}-s^2+y^2\right)\pm\sqrt{\left(\frac{a-b}{u}-s^2+y^2\right)^2+4\left(s\,y-c\right)^2}}{2\left(s\,y-c\right)}. $$
I'll leave it to you to conclude.