Sine of Integer Multiple of Argument

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Theorem

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

Formulation 1

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


Formulation 2

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


Formulation 3

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


Formulation 4

\(\ds \map \sin {n \theta}\) \(=\) \(\ds \paren {2 \cos \theta } \map \sin {\paren {n - 1 } \theta} - \map \sin {\paren {n - 2 } \theta}\)


Formulation 5

\(\ds \sin n \theta\) \(=\) \(\ds \paren {\sin \frac {n \pi} 2} \paren {\sin \theta} + \paren {2 \cos \theta} \paren {\map \sin {\paren {n - 1} \theta} - \map \sin {\paren {n - 3} \theta} + \map \sin {\paren {n - 5} \theta} - \cdots}\)
\(\ds \) \(=\) \(\ds \paren {\sin \frac {n \pi} 2} \paren {\sin \theta} + 2 \cos \theta \paren {\sum_{k \mathop = 0}^n \paren {\sin \frac {k \pi} 2} \map \sin {\paren {n - k} \theta} }\)


Formulation 6

\(\ds \map \sin {n \theta}\) \(=\) \(\ds \paren {2 \sin \theta } \map \cos {\paren {n - 1 } \theta} + \map \sin {\paren {n - 2 } \theta}\)


Formulation 7

\(\ds \sin n \theta\) \(=\) \(\ds \paren {\sin^2 \frac {n \pi} 2} \paren {\sin \theta} + \paren {2 \sin \theta } \paren {\paren {0} \map \cos {\paren {n - 0} \theta} + \paren {1} \map \cos {\paren {n - 1} \theta} + \paren {0} \map \cos {\paren {n - 2} \theta} + \paren {1} \map \cos {\paren {n - 3} \theta} + \paren {0} \map \cos {\paren {n - 4} \theta} + \paren {1} \map \cos {\paren {n - 5} \theta} - \cdots}\)
\(\ds \) \(=\) \(\ds \paren {\sin^2 \frac {n \pi} 2} \paren {\sin \theta} + 2 \sin \theta \paren {\sum_{k \mathop = 0}^{n - 1} \paren {\sin^2 \frac {k \pi} 2} \map \cos {\paren {n - k} \theta} }\)


Formulation 8

$\sin n \theta = \map \sin {\paren {n - 1 } \theta} \paren { a_0 - \cfrac 1 {a_1 - \cfrac 1 {a_2 - \cfrac 1 {\ddots \cfrac {} {a_{n-3} - \cfrac 1 {a_{n-2}}} }}} }$

where $a_0 = a_1 = a_2 = \ldots = a_{n-2} = 2 \cos \theta$.


Formulation 9

$\sin n \theta = \map \cos {\paren {n - 1} \theta} \paren {a_0 + \cfrac 1 {a_1 + \cfrac 1 {a_2 + \cfrac 1 {\ddots \cfrac {} {a_{n - r} } } } } }$

where:

$r = \begin {cases} 2 & : \text {$n$ is even} \\ 1 & : \text {$n$ is odd} \end {cases}$
$a_k = \begin {cases} 2 \sin \theta & : \text {$k$ is even} \\ -2 \sin \theta & : \text {$k$ is odd and $k < n - 1$} \\ \sin \theta & : k = n - 1 \end {cases}$


Also see