Sine of Integer Multiple of Argument

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Theorem

For $n \in \Z_{>0}$:

Formulation 1

\(\displaystyle \sin n \theta\) \(=\) \(\displaystyle \sin \theta \paren {\paren {2 \cos \theta}^{n - 1} - \dbinom {n - 2} 1 \paren {2 \cos \theta}^{n - 3} + \dbinom {n - 3} 2 \paren {2 \cos \theta}^{n - 5} - \cdots}\)
\(\displaystyle \) \(=\) \(\displaystyle \sin \theta \paren {\sum_{k \mathop \ge 0} \paren {-1}^k \binom {n - \paren {k + 1} } k \paren {2 \cos \theta}^{n - \paren {2 k + 1} } }\)


Formulation 2

\(\displaystyle \sin n \theta\) \(=\) \(\displaystyle \cos^n \theta \paren {\paren {\tan \theta} - \dbinom n 3 \paren {\tan \theta}^3 + \dbinom n 5 \paren {\tan \theta}^5 - \cdots}\)
\(\displaystyle \) \(=\) \(\displaystyle \cos^n \theta \sum_{k \mathop \ge 0} \paren {-1}^k \dbinom n {2 k + 1} \paren {\tan^{2 k + 1} \theta}\)


Formulation 3

\(\displaystyle \sin n \theta\) \(=\) \(\displaystyle \sin \theta \cos^{n - 1} \theta \paren {1 + 1 + \frac {\cos 2 \theta} {\cos^2 \theta} + \frac {\cos 3 \theta} {\cos^3 \theta} + \cdots + \frac {\cos \paren {n - 1} \theta} {\cos^{n - 1} \theta} }\)
\(\displaystyle \) \(=\) \(\displaystyle \sin \theta \cos^{n - 1} \theta \sum_{k \mathop = 0}^{n - 1} \frac {\cos k \theta} {\cos^k \theta}\)


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