Category:Examples of Sine of Integer Multiple of Argument

From ProofWiki
Jump to navigation Jump to search

This category contains examples of use of Sine of Integer Multiple of Argument.

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

\(\ds \sin n \theta\) \(=\) \(\ds \paren {\sin \frac {n \pi} 2} \paren {\sin \theta} + \paren {2 \cos \theta} \paren {\map \sin {\paren {n - 1} \theta} - \map \sin {\paren {n - 3} \theta} + \map \sin {\paren {n - 5} \theta} - \cdots}\)
\(\ds \) \(=\) \(\ds \paren {\sin \frac {n \pi} 2} \paren {\sin \theta} + 2 \cos \theta \paren {\sum_{k \mathop = 0}^n \paren {\sin \frac {k \pi} 2} \map \sin {\paren {n - k} \theta} }\)
Retrieved from "https://proofwiki.org/w/index.php?title=Category:Examples_of_Sine_of_Integer_Multiple_of_Argument&oldid=510972"