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I wonder whether the sequence $s_{n} = \sum_{k = 1}^{n} \sin k$ is bounded.

The answer seems no, but I have no idea how to prove this from the irrationality of $\pi$.

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    $\begingroup$ It's not a research level question. You should ask it at math.stack.exchange. $\endgroup$
    – Davide Giraudo
    Jun 9, 2012 at 11:31

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In this particular case, use $s_n(\theta)=\sum_{k=0}^n\sin(k\theta)=\Im(\sum_{k=0}^n\exp(ki\theta))=\dots=\frac{\sin(n\theta/2)\sin((n+1)\theta/2)}{\sin (\theta/2)}$

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    $\begingroup$ In particular, $s_n$ is bounded. $\endgroup$
    – David E Speyer
    Jun 9, 2012 at 11:55

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