A Curious sum of sines and cosines with angles in arithmetic progression

Question

I am stuck with the following problem: \begin{align} \max_{\theta\in\mathbb{R}}\sum_{i=1}^{N}(a_i\sin(i\theta)+b_i\cos(i\theta)), \end{align} where for $i(1\leq i\leq N$)$, a_i$ and $b_i$ are real numbers. If a solution is not known, please point to references which deals with this sort of equations.

Answer 1

This is a trigonometric polynomial, or a finite instance of a Fourier series with period $2\pi$. There is nothing obvious that would help you to find the maximum, take the first derivative and look for its roots.