Cosine of Integer Multiple of Argument/Formulation 8/Examples/Cosine of Quintuple Angle
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Example of Use of Cosine of Integer Multiple of Argument/Formulation 8
- $\map \cos {5 \theta } = \map \cos {4 \theta} \paren { 2 \cos \theta - \cfrac 1 {2 \cos \theta - \cfrac 1 {2\cos \theta - \cfrac 1 {2 \cos \theta - \cfrac 1 {\cos \theta } } }} }$
Proof
Follows directly from the Cosine of Integer Multiple of Argument: Formulation 8:
Explicit derivation illustrated below:
\(\ds \map \cos {5 \theta}\) | \(=\) | \(\ds \paren {2 \cos \theta } \map \cos {4 \theta} - \map \cos {3 \theta}\) | Cosine of Integer Multiple of Argument: Formulation 4 | |||||||||||
\(\ds \) | \(=\) | \(\ds \map \cos {4 \theta} \paren { \paren {2 \cos \theta } - \frac {\map \cos {3 \theta} } {\map \cos {4 \theta} } }\) | Factor out $\map \cos {4 \theta}$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \map \cos {4 \theta} \paren { 2 \cos \theta - \cfrac 1 {\cfrac {\map \cos {4 \theta} } {\map \cos {3 \theta} } } }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \map \cos {4 \theta} \paren { 2 \cos \theta - \cfrac 1 {\cfrac {\paren {2 \cos \theta } \map \cos {3 \theta} - \map \cos {2 \theta} } {\map \cos {3 \theta} } } }\) | Cosine of Integer Multiple of Argument: Formulation 4 | |||||||||||
\(\ds \) | \(=\) | \(\ds \map \cos {4 \theta} \paren { 2 \cos \theta - \cfrac 1 {2 \cos \theta - \cfrac {\map \cos {2 \theta} } {\map \cos {3 \theta} } } }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \map \cos {4 \theta} \paren { 2 \cos \theta - \cfrac 1 {2 \cos \theta - \cfrac 1 {\cfrac {\map \cos {3 \theta} } {\map \cos {2 \theta} } } } }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \map \cos {4 \theta} \paren { 2 \cos \theta - \cfrac 1 {2 \cos \theta - \cfrac 1 {\cfrac {\paren {2 \cos \theta } \map \cos {2 \theta} - \map \cos { \theta} } {\map \cos {2 \theta} } } } }\) | Cosine of Integer Multiple of Argument: Formulation 4 | |||||||||||
\(\ds \) | \(=\) | \(\ds \map \cos {4 \theta} \paren { 2 \cos \theta - \cfrac 1 {2 \cos \theta - \cfrac 1 {2 \cos \theta - \cfrac {\map \cos { \theta} } {\map \cos {2 \theta} } } } }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \map \cos {4 \theta} \paren { 2 \cos \theta - \cfrac 1 {2 \cos \theta - \cfrac 1 {2 \cos \theta - \cfrac 1 {\cfrac {\map \cos {2 \theta} } {\map \cos {\theta} } } } } }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \map \cos {4 \theta} \paren {2 \cos \theta - \cfrac 1 {2 \cos \theta - \cfrac 1 {2 \cos \theta - \cfrac 1 {\cfrac {\paren {2 \cos \theta } \map \cos {\theta} - \map \cos {0 } } {\map \cos {\theta} } } } } }\) | Cosine of Integer Multiple of Argument: Formulation 4 | |||||||||||
\(\ds \) | \(=\) | \(\ds \map \cos {4 \theta} \paren {2 \cos \theta - \cfrac 1 {2 \cos \theta - \cfrac 1 {2 \cos \theta - \cfrac 1 {2 \cos \theta - \cfrac 1 {\cos \theta } } } } }\) |
$\blacksquare$