1

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 $S(x)$.

J. M. ain't a mathematician
  • 75,051
Alexander51413
  • 456
  • 5
  • 18
  • Something similar https://math.stackexchange.com/questions/2618462/proving-complex-series-1-cos-theta-cos2-theta-cos-n-theta/ – rtybase Mar 30 '18 at 00:40

1 Answers1

2

HINT: $e^{i\varphi}=\cos\varphi+i\sin\varphi$. Then compute $$ \sum_{k=1}^{x}e^{ik} $$ and take the imaginary part.

Przemysław Scherwentke
  • 13,668
  • 5
  • 35
  • 56