Sum of Sines of Arithmetic Sequence of Angles
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
\(\ds \sum_{k \mathop = 1}^n \map \sin {\theta + k \alpha}\) | \(=\) | \(\ds \map \sin {\theta + \alpha} + \map \sin {\theta + 2 \alpha} + \map \sin {\theta + 3 \alpha} + \dotsb\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \map \sin {\theta + \frac {n + 1} 2 \alpha}\frac {\map \sin {n \alpha / 2} } {\map \sin {\alpha / 2} }\) |