Cosine of Integer Multiple of Argument/Formulation 5/Examples/Cosine of Quintuple Angle
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Example of Use of Cosine of Integer Multiple of Argument: Formulation 5
- $\cos 5 \theta = \cos \theta + 2 \cos \theta \paren {\cos 4 \theta - \cos 2 \theta}$
Proof
Follows directly from Cosine of Integer Multiple of Argument: Formulation 5:
Explicit derivation illustrated below:
\(\ds \cos 5 \theta\) | \(=\) | \(\ds 2 \cos \theta \cos 4 \theta - \cos 3 \theta\) | Cosine of Integer Multiple of Argument: Formulation 4 | |||||||||||
\(\ds \cos 3 \theta\) | \(=\) | \(\ds 2 \cos \theta \cos 2 \theta - \cos \theta\) | Cosine of Integer Multiple of Argument: Formulation 4 | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds \cos 5 \theta\) | \(=\) | \(\ds 2 \cos \theta \paren {\cos 4 \theta - \cos 2 \theta} + \cos \theta\) | |||||||||||
\(\ds \) | \(=\) | \(\ds \cos \theta + 2 \cos \theta \paren {\cos 4 \theta - \cos 2 \theta}\) |
$\blacksquare$