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