I would like to solve the following system of equations for $α_1$ and $α_2$:
$$ \begin{bmatrix} \frac{\alpha_1 sin(\alpha_1 +\alpha_2)-sin(\alpha_1) \alpha_1 +sin(\alpha_1) \alpha_2}{\alpha_1 \alpha_2}\\ \frac{-\alpha_1 cos(\alpha_1+\alpha_2)+cos(\alpha_1)\alpha_1+\alpha_2-\alpha_2 cos(\alpha_1)}{\alpha_1 \alpha_2} \end{bmatrix} = \begin{bmatrix} x\\ y \end{bmatrix} $$
Obviously, this is not straight forward. The equations describe a Cartesian robotic end-effector position depending on the two angles $α_1$ and $α_2$. I would like to get your feedback how to proceed.
1.) Is there an analytic solution to compute $α_1$ and $α_2$ for a given $x$ and $y$?
2.) If not, would you try to fit an approximation around the working point?
3.) Is there any other approach I don't have in mind?
Every feedback is appreciated. Thank you very much!