Category:Examples of Cosine of Integer Multiple of Argument

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

Formulation 1

\(\ds \cos n \theta\) \(=\) \(\ds \dfrac 1 2 \paren {\paren {2 \cos \theta}^n - \dfrac n 1 \paren {2 \cos \theta}^{n - 2} + \dfrac n 2 \dbinom {n - 3} 1 \paren {2 \cos \theta}^{n - 4} - \dfrac n 3 \dbinom {n - 4} 2 \paren {2 \cos \theta}^{n - 6} + \cdots}\)
\(\ds \) \(=\) \(\ds \dfrac 1 2 \paren {\paren {2 \cos \theta}^n + \sum_{k \mathop \ge 1} \paren {-1}^k \dfrac n k \dbinom {n - \paren {k + 1} } {k - 1} \paren {2 \cos \theta}^{n - 2 k} }\)


Formulation 2

\(\ds \cos n \theta\) \(=\) \(\ds \cos^n \theta \paren {1 - \dbinom n 2 \paren {\tan \theta}^2 + \dbinom n 4 \paren {\tan \theta}^4 - \cdots}\)
\(\ds \) \(=\) \(\ds \cos^n \theta \sum_{k \mathop \ge 0} \paren {-1}^k \dbinom n {2 k } \paren {\tan^{2 k } \theta}\)


Formulation 3

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


Formulation 4

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


Formulation 5

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