I have two equations as follows: $$ \left\{ \begin{array}{c} (\Delta_{11}\cos(\alpha) + \Delta_{12})\cos(\theta) + (\Delta_{21}\cos(\alpha) + \Delta_{22})\sin(\theta) = \Delta_{31}\sin(\alpha) + \Delta_{32} \\ (\Omega_{11}\cos(\alpha) + \Omega_{12})\cos(\theta) + (\Omega_{21}\cos(\alpha) + \Omega_{22})\sin(\theta) = \Omega_{31}\sin(\alpha) + \Omega_{32} \\ \end{array} \right. $$ where $\Delta_{i,j}$ and $\Omega_{i,j}$ are known scalar values.
I want to ask if there is any solution for $\theta$ and $\alpha$? Both are also scalars.
Thank you very much for your help!
What I have tried so far:
1. One possible solution is to convert sin, cos to half angle tan, but this leads to an 8th order polynomial. Would it be possible to solve this polynomial for $\alpha$ and $\theta$ though?
2. Another special case, is when both $Δ_{32}$ and $Ω_{32}$ are zeros. After eliminating sin(α), the equation reduces to a 4th order polynomial, then it is in closed form.
3. Does anyone think it would be possible to reduce the equations to a 4th order polynomial for the general case? (i.e. when $Δ_{32}$ and $Ω_{32}$ are not zeros).