# Is there a good lower bound for $|\sin(k\theta)|$ when $e^{i\theta}$ is algebraic and $\theta/\pi$ is irrational?

The precise question is: given $$\lambda>1$$, does there always exist $$C>0$$ such that $$|\sin(k\theta)|>C\lambda^{-k}$$ for all $$k$$?

My original purpose is to show that: whenever $$\lambda>1$$, and $$\theta$$ is given as in the title, one has $$\lim_{k\to\infty}\lambda^k|\sin(k\theta)|=+\infty$$.

Thanks to everyone!