Cosine of Integer Multiple of Argument/Formulation 9/Examples/Cosine of Quintuple Angle

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Example of Use of Cosine of Integer Multiple of Argument/Formulation 9

$\map \cos {5 \theta } = \map \sin {4 \theta} \paren { -2 \sin \theta + \cfrac 1 {2 \sin \theta + \cfrac 1 {-2\sin \theta + \cfrac 1 {2 \sin \theta } }} }$


Proof

Follows directly from the Cosine of Integer Multiple of Argument: Formulation 9:

Explicit derivation illustrated below:

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

$\blacksquare$

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