Bounds for $S_N(\alpha) = \sum_{n=1}^N \sin(\alpha n)$

Question

Let $\alpha \in \mathbb{R}$ and $N$ be a positive integer. Are there any good bounds known for the absolute value of the sum $$ S_N(\alpha) = \sum_{n=1}^N \sin(\alpha n) $$ for $N$ large and which are uniform in $\alpha$? That is, what can we say about $\|S_N\|_{\infty}$? Clearly $\|S_N\|_\infty \leq N$. But can we do better? Thanks.

Answer 1

HINT

Notice that \begin{align*} S_{N}(\alpha) = \sum_{n=1}^{N}\sin(\alpha n) = \mathrm{Im}\sum_{n=1}^{N}e^{i\alpha n} = \mathrm{Im}\left[\frac{e^{i\alpha} - e^{i(N+1)\alpha}}{1 - e^{i\alpha}}\right] \end{align*}