Sine of Integer Multiple of Argument/Formulation 2/Examples/Sine of Quintuple Angle
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Example of Use of Sine of Integer Multiple of Argument: Formulation 2
- $\sin 5 \theta = \cos^5 \theta \paren {5 \tan \theta - 10 \tan^3 \theta + \tan^5 \theta}$
Proof
\(\ds \sin 5 \theta\) | \(=\) | \(\ds \cos^n \theta \paren {\dbinom n 1 \tan \theta - \dbinom n 3 \paren {\tan \theta}^3 + \dbinom n 5 \paren {\tan \theta}^5 - \cdots}\) | Sine of Integer Multiple of Argument: Formulation 2 | |||||||||||
\(\ds \) | \(=\) | \(\ds \cos^5 \theta \paren {\dbinom 5 1 \tan \theta - \dbinom 5 3 \paren {\tan \theta}^3 + \dbinom 5 5 \paren {\tan \theta}^5 }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \cos^5 \theta \paren {5 \tan \theta - 10 \tan^3 \theta + \tan^5 \theta}\) |
$\blacksquare$