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 | ||||||||||
\(\ds \) | \(=\) | \(\ds 2 \sin \theta \paren {\cos 5 \theta + \cos 3 \theta + \cos \theta}\) |
$\blacksquare$