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

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

$\sin 6 \theta = \sin \theta \paren { \paren {2 \cos \theta}^5 - 4 \paren {2 \cos \theta}^3 + 3 \paren {2 \cos \theta} }$

$\ds \sin 6 \theta$

$=$

$\ds \sin \theta \paren {\sum_{k \mathop \ge 0} \paren {-1}^k \binom {6 - \paren {k + 1} } k \paren {2 \cos \theta}^{6 - \paren {2 k + 1} } }$

$\ds $

$=$

$\ds \sin \theta \paren { \binom 5 0 \paren {2 \cos \theta}^5 - \binom 4 1 \paren {2 \cos \theta}^3 + \binom 3 2 \paren {2 \cos \theta} }$

$\ds $

$=$

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

$\blacksquare$