Lagrange's Trigonometric Identities/Sine/Cosine Form
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Theorem
- $\ds \sum_{k \mathop = 0}^n \sin k \theta = \dfrac {\map \cos {\frac 1 2 \theta} - \map \cos {n \theta + \frac 1 2 \theta} } {2 \map \sin {\frac 1 2 \theta} }$
Proof
| \(\ds \map \sin {\alpha} \map \sin {\beta}\) | \(=\) | \(\ds \dfrac {\map \cos {\alpha - \beta} - \map \cos {\alpha + \beta} } 2\) | Werner Formula for Sine by Sine | |||||||||||
| \(\ds 2 \map \sin {\beta} \map \sin {\alpha}\) | \(=\) | \(\ds \map \cos {\alpha - \beta} - \map \cos {\alpha + \beta}\) | rearranging | |||||||||||
| \(\ds 2 \map \sin {\frac 1 2 \theta} \cdot \map \sin {k \theta}\) | \(=\) | \(\ds \map \cos {k \theta - \frac 1 2 \theta} - \map \cos {k \theta + \frac 1 2 \theta}\) | setting $\alpha = k \theta$ and $\beta = \frac 1 2 \theta$ | |||||||||||
| \(\ds 2 \map \sin {\frac 1 2 \theta} \cdot \sum_{k \mathop = 0}^n \map \sin {k \theta}\) | \(=\) | \(\ds \sum_{k \mathop = 0}^n \map \cos {k \theta - \frac 1 2 \theta} - \map \cos {k \theta + \frac 1 2 \theta}\) | Summing both sides |
The right hand side is a telescoping sum.
| \(\ds 2 \map \sin {\frac 1 2 \theta} \cdot \sum_{k \mathop = 0}^n \map \sin {k \theta}\) | \(=\) | \(\ds \map \cos {\frac {-1} 2 \theta} - \map \cos {n \theta + \frac 1 2 \theta}\) | only two terms survive | |||||||||||
| \(\ds 2 \map \sin {\frac 1 2 \theta} \cdot \sum_{k \mathop = 0}^n \map \sin {k \theta}\) | \(=\) | \(\ds \map \cos {\frac 1 2 \theta} - \map \cos {n \theta + \frac 1 2 \theta}\) | Cosine Function is Even | |||||||||||
| \(\ds \sum_{k \mathop = 0}^n \map \sin {k \theta}\) | \(=\) | \(\ds \frac {\map \cos {\frac 1 2 \theta} - \map \cos {n \theta + \frac 1 2 \theta} }{ 2 \map \sin {\frac 1 2 \theta} }\) | rearranging |
$\blacksquare$