Simplify $\sin(\alpha)+\sin(\alpha+x)+\sin(\alpha+2x)+\sin(\alpha+3x)+ \dotsb +\sin(\alpha+nx)$

Question

$n$ is known, $\alpha$ is irrelevant, and you are looking for $x$? How do you simplify:

$$\sin(\alpha)+\sin(\alpha+x)+\sin(\alpha+2x)+\sin(\alpha+3x)+ \dotsb +\sin(\alpha+nx)$$

Answer 1

Hint

Denoting by $S$ your sum as $\sin(\theta)=\Im(e^{i \theta})$ you have: $$S=\Im \left( \sum_{k=0}^n e^{i(\alpha +k x)} \right)= \Im\left(e^{i\alpha} \sum_{k=0}^n (e^{ix})^k \right)$$ You can then use the formula for a sum of a geometric sequence to have an explicit formula.