Cosine of Sum/Proof 6
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Theorem
- $\map \cos {a + b} = \cos a \cos b - \sin a \sin b$
Proof
| \(\ds \map \cos {a + b}\) | \(=\) | \(\ds \map \cos {a - \paren {-b} }\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \cos a \map \cos {-b} + \sin a \map \sin {-b}\) | Cosine of Difference | |||||||||||
| \(\ds \) | \(=\) | \(\ds \cos a \cos b + \sin a \paren {-\sin b}\) | Cosine Function is Even, Sine Function is Odd | |||||||||||
| \(\ds \) | \(=\) | \(\ds \cos a \cos b - \sin a \sin b\) | simplifying |
$\blacksquare$
Sources
- 1953: L. Harwood Clarke: A Note Book in Pure Mathematics ... (previous) ... (next): $\text V$. Trigonometry: The addition formulae