Sum of Complex Exponentials of i times Arithmetic Sequence of Angles

Theorem

Let $\alpha \in \R$ be a real number such that $\alpha \ne 2 \pi k$ for $k \in \Z$.

Then:

Formulation 1

$\ds \sum_{k \mathop = 0}^n e^{i \paren {\theta + k \alpha} } = \paren {\map \cos {\theta + \frac {n \alpha} 2} + i \map \sin {\theta + \frac {n \alpha} 2} } \frac {\map \sin {\alpha \paren {n + 1} / 2} } {\map \sin {\alpha / 2} }$

Formulation 2

$\ds \sum_{k \mathop = 1}^n e^{i \paren {\theta + k \alpha} } = \paren {\map \cos {\theta + \frac {n + 1} 2 \alpha} + i \map \sin {\theta + \frac {n + 1} 2 \alpha} } \frac {\map \sin {n \alpha / 2} } {\map \sin {\alpha / 2} }$

Formulation 3

$\ds \sum_{k \mathop = p}^q e^{i \paren {\theta + k \alpha} } = \paren {\map \cos {\theta + \frac {\paren {p + q} \alpha} 2} + i \map \sin {\theta + \frac {\paren {p + q} \alpha} 2} } \frac {\map \sin {\paren {q - p + 1} \alpha / 2} } {\map \sin {\alpha / 2} }$