Sine of Integer Multiple of Argument/Formulation 1/Examples/Sine of Quintuple Angle

Example of Use of Sine of Integer Multiple of Argument: Formulation 1

$\sin 5 \theta = \sin \theta \paren { \paren {2 \cos \theta}^4 - 3 \paren {2 \cos \theta}^2 + 1 }$

$\ds \sin 5 \theta$

$=$

$\ds \sin \theta \paren {\sum_{k \mathop \ge 0} \paren {-1}^k \binom {5 - \paren {k + 1} } k \paren {2 \cos \theta}^{5 - \paren {2 k + 1} } }$

$\ds $

$=$

$\ds \sin \theta \paren { \binom 4 0 \paren {2 \cos \theta}^4 - \binom 3 1 \paren {2 \cos \theta}^2 + \binom 2 2 }$

$\ds $

$=$

$\ds \sin \theta \paren { \paren {2 \cos \theta}^4 - 3 \paren {2 \cos \theta}^2 + 1 }$

$\blacksquare$