Cosine of Integer Multiple of Argument/Formulation 7/Examples/Cosine of Sextuple Angle

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Example of Use of Cosine of Integer Multiple of Argument/Formulation 7

$\map \cos {6 \theta } = 1 - 2 \sin \theta \paren {\sin 5 \theta + \sin 3 \theta + \sin \theta}$


Proof

Follows directly from the Cosine of Integer Multiple of Argument/Formulation 7:

Explicit derivation illustrated below:

\(\ds \map \cos {6 \theta}\) \(=\) \(\ds -2 \sin \theta \map \sin {5 \theta} + \map \cos {4 \theta}\) Cosine of Integer Multiple of Argument: Formulation 6
\(\ds \map \cos {4 \theta}\) \(=\) \(\ds -2 \sin \theta \map \sin {3 \theta} + \map \cos {2 \theta}\) Cosine of Integer Multiple of Argument: Formulation 6
\(\ds \map \cos {6 \theta}\) \(=\) \(\ds -2 \sin \theta \map \sin {5 \theta} - 2 \sin \theta \map \sin {3 \theta} - 2 \sin \theta \sin \theta + 1\) Double Angle Formula for Cosine: Corollary $2$
\(\ds \) \(=\) \(\ds 1 - 2 \sin \theta \paren {\sin 5 \theta + \sin 3 \theta + \sin \theta}\)

$\blacksquare$