To put it more rigorously: Does there exist constants $0 < a < 1$ and $0 < b < 1$ such that for all real $\theta$ which are non-rational multiples of $\pi$ and all natural numbers $n$ it holds that $b a^n\leq \sin(\theta n) $. My reasoning is that because $\theta$ is restricted to be a non-rational multiple of $\pi$, we can't find such constants as $\sin$ gets arbitrarily close to 0. But, I can't find a proof. Are there any related theorems that might help me?
Update: I changed the question so that $\theta$ has to be a non-rational multiple of $\pi$, to make it more interesting. I stumbled upon this problem in a different context, and made a mistake when reformulating it here.
Another alternative would be to modify the inequality as $b a^n\leq y \sin(\theta n) + x$ with the restriction that $x\geq 2y > 0$.