According to Wolfram Alpha: $$\sum\limits_{k=1}^x\sin(k) = \frac{\sin(x)-\cot(\frac12)\cos(x)+\cot(\frac12)}2$$ I know I should show some attempt at proving this problem, however I have no idea where to start at all. The only idea I have is setting $S(x) = \sin(1) + \sin(2) + \sin(3) + {...} + \sin(x)$ and then manipulating it somehow to solve for $S(x)$.
2026-04-03 02:33:14.1775183594
Evaluating $\sum\limits_{k=1}^x\sin(k)$
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HINT: $e^{i\varphi}=\cos\varphi+i\sin\varphi$. Then compute $$ \sum_{k=1}^{x}e^{ik} $$ and take the imaginary part.