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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!

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