Sine of Sum of Three Angles

Theorem

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


Proof

\(\ds \map \sin {A + B + C}\) \(=\) \(\ds \map \sin {A + B} \cos C + \map \cos {A + B} \sin C\) Sine of Sum
\(\ds \) \(=\) \(\ds \paren {\sin A \cos B + \cos A \sin B} \cos C + \paren {\cos A \cos B - \sin A \sin B} \sin C\) Sine of Sum, Cosine of Sum
\(\ds \) \(=\) \(\ds \sin A \cos B \cos C + \cos A \sin B \cos C + \cos A \cos B \sin C - \sin A \sin B \sin C\) multiplying out

$\blacksquare$

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