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

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

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


Proof

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

Explicit derivation illustrated below:

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

$\blacksquare$

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