Sum of Sines of Arithmetic Sequence of Angles

From ProofWiki
Jump to navigation Jump to search

Theorem

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

Then:

Formulation 1

\(\ds \sum_{k \mathop = 0}^n \map \sin {\theta + k \alpha}\) \(=\) \(\ds \sin \theta + \map \sin {\theta + \alpha} + \map \sin {\theta + 2 \alpha} + \map \sin {\theta + 3 \alpha} + \dotsb\)
\(\ds \) \(=\) \(\ds \frac {\map \sin {\alpha \paren {n + 1} / 2} } {\map \sin {\alpha / 2} } \map \sin {\theta + \frac {n \alpha} 2}\)


Formulation 2

\(\ds \sum_{k \mathop = 1}^n \map \sin {\theta + k \alpha}\) \(=\) \(\ds \map \sin {\theta + \alpha} + \map \sin {\theta + 2 \alpha} + \map \sin {\theta + 3 \alpha} + \dotsb\)
\(\ds \) \(=\) \(\ds \map \sin {\theta + \frac {n + 1} 2 \alpha}\frac {\map \sin {n \alpha / 2} } {\map \sin {\alpha / 2} }\)


Also see