Sine of Integer Multiple of Argument/Formulation 2/Examples/Sine of Sextuple Angle

From ProofWiki
Jump to navigation Jump to search

Example of Use of Sine of Integer Multiple of Argument/Formulation 2

$\map \sin {6 \theta } = \cos^6 \theta \paren {6 \tan \theta - 20 \tan^3 \theta + 6 \tan^5 \theta}$


Proof

Follows directly from the Sine of Integer Multiple of Argument/Formulation 2:

\(\ds \map \sin {6 \theta}\) \(=\) \(\ds \cos^n \theta \paren {\dbinom n 1 \tan \theta - \dbinom n 3 \paren {\tan \theta}^3 + \dbinom n 5 \paren {\tan \theta}^5 - \cdots}\) Sine of Integer Multiple of Argument/Formulation 2
\(\ds \) \(=\) \(\ds \cos^6 \theta \paren {\dbinom 6 1 \tan \theta - \dbinom 6 3 \paren {\tan \theta}^3 + \dbinom 6 5 \paren {\tan \theta}^5 }\)
\(\ds \) \(=\) \(\ds \cos^6 \theta \paren {6 \tan \theta - 20 \tan^3 \theta + 6 \tan^5 \theta }\)


$\blacksquare$