I would like to calculate the Riemann sum of $\sin(x)$. Fun starts here:
$R = \frac{\pi}{n} \sum_{j=1}^n \sin\left(\frac{\pi}{n}\cdot j\right)$
What would be the simplest way to calculate the sum of $\sin\left(\frac{\pi}{n}\cdot j\right)$, so that one could proceed to evaluating the limit and thus getting the value of the Riemann sum, in other words - the integral?
There maybe a way using $\mathbb{C}$?