Sine of Sum of Three Angles

Theorem

$\map \sin {A + B + C} = \sin A \cos B \cos C + \cos A \sin B \cos C + \cos A \cos B \sin C - \sin A \sin B \sin C$

$\ds \map \sin {A + B + C}$

$=$

$\ds \map \sin {A + B} \cos C + \map \cos {A + B} \sin C$

$\ds $

$=$

$\ds \paren {\sin A \cos B + \cos A \sin B} \cos C + \paren {\cos A \cos B - \sin A \sin B} \sin C$

$\ds $

$=$

$\ds \sin A \cos B \cos C + \cos A \sin B \cos C + \cos A \cos B \sin C - \sin A \sin B \sin C$

multiplying out

$\blacksquare$