I would like to know if there is an analytical way to find the roots of series of sine functions given by $$ \sum_{n=0}^N \sin \left( \omega_n t \right) = 0 $$ Here, $\omega_n = \frac{1}{h}\textrm{arccot}\left(\frac{4-h^2\alpha_n^2}{4h\alpha_n}\right) - \alpha_n$, where $\alpha_n$ doesn't necessarily have any restrictions on it. For simple sequences of $\omega_n$, WolframAlpha seems to provide exact solutions for examples [e.g. this] I've fed it but does not cite any methodology as to how it got to the answer. I do know that I could solve for these roots using some numerical root finding method (bisection, Newton's, Secant, etc) but I'm concerned with whether or not there exists an analytical way to do this.
EDIT: Added so info for $\omega_n$.