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

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

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


Proof

Follows directly from the Sine of Integer Multiple of Argument: Formulation 7:

Explicit derivation illustrated below:

\(\ds \sin 6 \theta\) \(=\) \(\ds \paren {2 \sin \theta} \cos 5 \theta + \sin 4 \theta\) Sine of Integer Multiple of Argument: Formulation 6
\(\ds \sin 4 \theta\) \(=\) \(\ds \paren {2 \sin \theta} \cos 3 \theta + \sin 2 \theta\) Sine of Integer Multiple of Argument: Formulation 6
\(\ds \leadsto \ \ \) \(\ds \sin 6 \theta\) \(=\) \(\ds \paren {2 \sin \theta} \cos 5 \theta + \paren {2 \sin \theta} \cos 3 \theta + \paren {2 \sin \theta} \cos \theta\) Double Angle Formula for Sine
\(\ds \) \(=\) \(\ds 2 \sin \theta \paren {\cos 5 \theta + \cos 3 \theta + \cos \theta}\)

$\blacksquare$