Sum of Complex Exponentials of i times Arithmetic Sequence of Angles/Formulation 3

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

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

Proof
First note that if $\alpha = 2 \pi k$ for $k \in \Z$, then $e^{i \alpha} = 1$.