What is the formula for: $$\sum_{i=1}^n \sin(i\theta)$$
I looked at various websites but cant find it. Does such a formula exist?
What is the formula for: $$\sum_{i=1}^n \sin(i\theta)$$
I looked at various websites but cant find it. Does such a formula exist?
Hint
Notice that $$\sin(k\theta)=\operatorname{Im}e^{ik\theta}$$ and use the geometric sum.
$\displaystyle\sum_{i=1}^{n}\sin (n\theta)$
=$Im(e^{in\theta})$
=$\displaystyle\sum_{i=1}^{n}Im(1+in\theta+\frac{(n\theta)^{2}}{2}-i\frac{(n\theta)^{3}}{6}+\frac{(n\theta)^{4}}{24}-i\frac{(n\theta)^{5}}{120}+...)$
$\displaystyle=\sum_{i=1}^{n}(n\theta-\frac{(n\theta)^{3}}{6}-\frac{(n\theta)^{5}}{120}-...)$