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

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

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


Proof

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

Explicit derivation illustrated below:

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

$\blacksquare$

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