Cosine of Sum of Three Angles
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Theorem
- $\map \cos {A + B + C} = \cos A \cos B \cos C - \sin A \sin B \cos C - \sin A \cos B \sin C - \cos A \sin B \sin C$
Proof
| \(\ds \map \cos {A + B + C}\) | \(=\) | \(\ds \map \cos {A + B} \cos C - \map \sin {A + B} \sin C\) | Cosine of Sum | |||||||||||
| \(\ds \) | \(=\) | \(\ds \paren {\cos A \cos B - \sin A \sin B} \cos C - \paren {\sin A \cos B + \cos A \sin B} \sin C\) | Cosine of Sum, Sine of Sum | |||||||||||
| \(\ds \) | \(=\) | \(\ds \cos A \cos B \cos C - \sin A \sin B \cos C - \sin A \cos B \sin C - \cos A \sin B \sin C\) | multiplying out |
$\blacksquare$