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