Consider a scaled sine function, $\sin(2\pi x/2^n)$, for some positive integer $n$. For this, I have the following linear combination.

$$ \sum_{x=1}^{2^{n-2}} c_x \sin(2\pi x/2^n).$$ (The upper limit to the sum is $2^{n-2}$.)

The question is whether there exist $c_x \in \{0, \pm 1, \pm 2\}$, not all $0$, that make the above expression $0$, for infinitely many $n$?

If it helps, the above came up in a computation concerning the discrete Fourier Transform.