Sine of Integer Multiple of Argument/Formulation 6
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Theorem
For $n \in \Z$:
| \(\ds \map \sin {n \theta}\) | \(=\) | \(\ds \paren {2 \sin \theta } \map \cos {\paren {n - 1 } \theta} + \map \sin {\paren {n - 2 } \theta}\) |
Proof
| \(\ds \map \sin {n \theta}\) | \(=\) | \(\ds \map \sin {\paren {n - 1 } \theta + \theta }\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \sin \theta \map \cos {\paren {n - 1 } \theta} + \cos \theta \map \sin {\paren {n - 1 } \theta}\) | Sine of Sum | |||||||||||
| \(\ds \) | \(=\) | \(\ds \sin \theta \map \cos {\paren {n - 1 } \theta} + \cos \theta \paren { \sin \theta \map \cos {\paren {n - 2 } \theta} + \cos \theta \map \sin {\paren {n - 2 } \theta} }\) | Sine of Sum | |||||||||||
| \(\ds \) | \(=\) | \(\ds \sin \theta \map \cos {\paren {n - 1 } \theta} + \sin \theta \cos \theta \map \cos {\paren {n - 2 } \theta} + \cos^2 \theta \map \sin {\paren {n - 2 } \theta}\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \sin \theta \map \cos {\paren {n - 1 } \theta} + \sin \theta \cos \theta \map \cos {\paren {n - 2 } \theta} + \paren {1 - \sin^2 \theta } \map \sin {\paren {n - 2 } \theta}\) | Sum of Squares of Sine and Cosine | |||||||||||
| \(\ds \) | \(=\) | \(\ds \sin \theta \map \cos {\paren {n - 1 } \theta} + \sin \theta \paren { \cos \theta \map \cos {\paren {n - 2 } \theta} - \sin \theta \map \sin {\paren {n - 2 } \theta} } + \map \sin {\paren {n - 2 } \theta}\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \sin \theta \map \cos {\paren {n - 1 } \theta} + \sin \theta \map \cos {\paren {n - 1 } \theta} + \map \sin {\paren {n - 2 } \theta}\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \paren {2 \sin \theta } \map \cos {\paren {n - 1 } \theta} + \map \sin {\paren {n - 2 } \theta}\) |
$\blacksquare$