Can we solve for $c$ in the equation $\sum\limits_{i=0}^{N-1} \exp\left(-jc\sin\left(\frac{2\pi i}{N}-\frac{\pi k}{N}\right)\right)=0$?

Question

This question was posted on stackexchange, but with no response. So, I thought it appropriate to post it here too. Let $N\geq 1$ and $0\leq k\leq N-1$ be fixed numbers, and $c>0$ be unknown. Suppose we have

\begin{eqnarray} \sum\limits_{i=0}^{N-1} \exp\left(-jc\sin\left(\frac{2\pi i}{N}-\frac{\pi k}{N}\right)\right)=0, \end{eqnarray} where $j:=\sqrt{-1}$. Equivalently, we have \begin{eqnarray} \sum\limits_{i=0}^{N-1} \cos\left(c\sin\left(\frac{2\pi i}{N}-\frac{\pi k}{N}\right)\right)=0,~~~~~~~~~~~~~~~\sum\limits_{i=0}^{N-1} \sin\left(c\sin\left(\frac{2\pi i}{N}-\frac{\pi k}{N}\right)\right)=0. \end{eqnarray}

Is it possible to simplify and solve the above equations to arrive at a closed form expression for $c$ in terms of $k$ and $N$?

Any help would be greatly appreciated.