leaving the context behind, I simplified a problem to the following equation:
solve for x: $$e^{\frac{-x}{a}} \cdot \sin(b + cx) = \sin(b)$$
Since my equations contains both sine and exponential function, I could neither use $\ln$ nor $\sin(x)^{-1}$. I tried using eulers identity to reduce all sine functions to exponential functions. Sadly, this couldn't help me either.
Thanks in advance.
You can rewrite this into $$ f(x) = e^{-x/a} \sin(b + cx) - \sin(a) $$ and look for the roots.
$f$ seems to be a variation of a sine function modulated by an exponential function.
E.g.
Looks like a signal from electrical engineering. Maybe they have a non-numerical solution?