Cosine of Sum/Corollary
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Corollary to Cosine of Sum
- $\map \cos {a - b} = \cos a \cos b + \sin a \sin b$
where $\sin$ denotes the sine and $\cos$ denotes the cosine.
Proof
| \(\displaystyle \map \cos {a - b}\) | \(=\) | \(\displaystyle \cos a \, \map \cos {-b} - \sin a \, \map \sin {-b}\) | Cosine of Sum | ||||||||||
| \(\displaystyle \) | \(=\) | \(\displaystyle \cos a \cos b - \sin a \, \map \sin {-b}\) | Cosine Function is Even | ||||||||||
| \(\displaystyle \) | \(=\) | \(\displaystyle \cos a \cos b + \sin a \sin b\) | Sine Function is Odd |
$\blacksquare$
Historical Note
The Cosine of Sum formula and its corollary were proved by François Viète in about $1579$.
Sources
- 1960: Walter Ledermann: Complex Numbers ... (previous) ... (next): $\S 4.5$. The Functions $e^z$, $\cos z$, $\sin z$
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 5$: Trigonometric Functions: $5.35$
- 1981: Murray R. Spiegel: Theory and Problems of Complex Variables (SI ed.) ... (previous) ... (next): $2$: Functions, Limits and Continuity: The Elementary Functions: $4$
- 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): Entry: compound angle formulae (in trigonometry)
