Cosine of Integer Multiple of Argument/Formulation 8/Examples/Cosine of Sextuple Angle

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Example of Use of Cosine of Integer Multiple of Argument/Formulation 8

$\cos 6 \theta = \cos 5 \theta \paren {2 \cos \theta - \cfrac 1 {2 \cos \theta - \cfrac 1 {2 \cos \theta - \cfrac 1 {2 \cos \theta - \cfrac 1 {2 \cos \theta - \cfrac 1 {\cos \theta} } } } } }$


Proof

Follows directly from the Cosine of Integer Multiple of Argument: Formulation 8:

Explicit derivation illustrated below:

\(\ds \sin 6 \theta\) \(=\) \(\ds \paren {2 \cos \theta} \cos 5 \theta - \cos 4 \theta\) Cosine of Integer Multiple of Argument: Formulation 4
\(\ds \) \(=\) \(\ds \cos 5 \theta \paren {2 \cos \theta - \frac {\cos 4 \theta} {\cos 5 \theta} }\) Factor out $\cos 5 \theta$
\(\ds \) \(=\) \(\ds \cos 5 \theta \paren {2 \cos \theta - \cfrac 1 {\cfrac {\cos 5 \theta} {\cos 4 \theta} } }\)
\(\ds \) \(=\) \(\ds \cos 5 \theta \paren {2 \cos \theta - \cfrac 1 {\cfrac {\paren {2 \cos \theta} \cos 4 \theta - \cos 3 \theta} {\cos 4 \theta} } }\) Cosine of Integer Multiple of Argument: Formulation 4
\(\ds \) \(=\) \(\ds \cos 5 \theta \paren {2 \cos \theta - \cfrac 1 {2 \cos \theta - \cfrac {\cos 3 \theta} {\cos 4 \theta} } }\)
\(\ds \) \(=\) \(\ds \cos 5 \theta \paren {2 \cos \theta - \cfrac 1 {2 \cos \theta - \cfrac 1 {\cfrac {\cos 4 \theta} {\cos 3 \theta} } } }\)
\(\ds \) \(=\) \(\ds \cos 5 \theta \paren {2 \cos \theta - \cfrac 1 {2 \cos \theta - \cfrac 1 {\cfrac {\paren {2 \cos \theta} \cos 3 \theta - \cos 2 \theta} {\cos 3 \theta} } } }\) Cosine of Integer Multiple of Argument: Formulation 4
\(\ds \) \(=\) \(\ds \cos 5 \theta \paren {2 \cos \theta - \cfrac 1 {2 \cos \theta - \cfrac 1 {2 \cos \theta - \cfrac {\cos 2 \theta} {\cos 3 \theta} } } }\)
\(\ds \) \(=\) \(\ds \cos 5 \theta \paren {2 \cos \theta - \cfrac 1 {2 \cos \theta - \cfrac 1 {2 \cos \theta - \cfrac 1 {\cfrac {\cos 3 \theta} {\cos 2 \theta} } } } }\)
\(\ds \) \(=\) \(\ds \cos 5 \theta \paren {2 \cos \theta - \cfrac 1 {2 \cos \theta - \cfrac 1 {2 \cos \theta - \cfrac 1 {\cfrac {\paren {2 \cos \theta} \cos 2 \theta - \cos \theta} {\cos 2 \theta} } } } }\) Cosine of Integer Multiple of Argument: Formulation 4
\(\ds \) \(=\) \(\ds \cos 5 \theta \paren {2 \cos \theta - \cfrac 1 {2 \cos \theta - \cfrac 1 {2 \cos \theta - \cfrac 1 {2 \cos \theta - \cfrac {\cos \theta} {\cos 2 \theta} } } } }\)
\(\ds \) \(=\) \(\ds \cos 5 \theta \paren {2 \cos \theta - \cfrac 1 {2 \cos \theta - \cfrac 1 {2 \cos \theta - \cfrac 1 {2 \cos \theta - \cfrac 1 {\cfrac {\cos 2 \theta} {\cos \theta} } } } } }\)
\(\ds \) \(=\) \(\ds \cos 5 \theta \paren {2 \cos \theta - \cfrac 1 {2 \cos \theta - \cfrac 1 {2 \cos \theta - \cfrac 1 {2 \cos \theta - \cfrac 1 {\cfrac {\paren {2 \cos \theta } \cos \theta - \cos 0} {\cos \theta} } } } } }\) Cosine of Integer Multiple of Argument: Formulation 4
\(\ds \) \(=\) \(\ds \cos 5 \theta \paren {2 \cos \theta - \cfrac 1 {2 \cos \theta - \cfrac 1 {2 \cos \theta - \cfrac 1 {2 \cos \theta - \cfrac 1 {2 \cos \theta - \cfrac 1 {\cos \theta} } } } } }\)

$\blacksquare$

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