Consider the following equation with variable $n \in \mathbb{N}$:$$\sum \limits_{i=1}^{k} \cos(n\theta_{i}) = 0.$$Given $\theta_1,\dots,\theta_k$, I'd like to determine whether there exists $n \in \mathbb{N}$ such that the above equation holds.For $k = 1$ a solution can be directly found in $O(1)$ time by taking the inverse cosine.For $k = 2$ we can first factor it using the identity$$\cos(A) + \cos(B) = 2\cos(\frac{A+B}{2})\cos(\frac{A-B}{2})$$and then set both terms individually to $0$. The running time will be $O(1)$.If all the $\theta_i$ are rational multiples of $\pi$ then the equation is periodic and solvable.Are there general algorithms of solving for $k > 2$ and for values of $\theta_i$ for which $\cos(\theta_i)$ are algebraic numbers? If not, then are there any undecidability results like for roots of polynomials (as mentioned in the paper below)?Undecidability in number theory.  Bjorn Poonen, Notices of the American Math Society, March 2008.I am considering all computation to be symbolic and exact. For example $\theta_1 = \cos^{-1}(1/3)$ is a possible input.
Complexity for finding zeroes of sum of cosines
algorithms;complexity theory;mathematical analysis