Cosine of Integer Multiple of Argument/Formulation 7/Examples/Cosine of Quintuple Angle
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Example of Use of Cosine of Integer Multiple of Argument: Formulation 7
- $\cos 5 \theta = \cos \theta - 2 \sin \theta \paren {\sin 4 \theta + \sin 2 \theta}$
Proof
Follows directly from the Cosine of Integer Multiple of Argument: Formulation 7:
Explicit derivation illustrated below:
\(\ds \map \cos {5 \theta}\) | \(=\) | \(\ds -2 \sin \theta \sin 4 \theta + \cos 3 \theta\) | Cosine of Integer Multiple of Argument: Formulation 6 | |||||||||||
\(\ds \map \cos {3 \theta}\) | \(=\) | \(\ds -2 \sin \theta \sin 2 \theta + \cos \theta\) | Cosine of Integer Multiple of Argument: Formulation 6 | |||||||||||
\(\ds \cos 5 \theta\) | \(=\) | \(\ds -2 \sin \theta \sin 4 \theta - 2 \sin \theta \sin 2 \theta + \cos \theta\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \cos \theta - 2 \sin \theta \paren {\sin 4 \theta + \sin 2 \theta}\) |
$\blacksquare$