Cosine of Difference
Theorem
- $\map \cos {a - b} = \cos a \cos b + \sin a \sin b$
where $\sin$ denotes the sine and $\cos$ denotes the cosine.
Proof 1
| \(\ds \map \cos {a - b}\) | \(=\) | \(\ds \cos a \map \cos {-b} - \sin a \map \sin {-b}\) | Cosine of Sum | |||||||||||
| \(\ds \) | \(=\) | \(\ds \cos a \cos b - \sin a \map \sin {-b}\) | Cosine Function is Even | |||||||||||
| \(\ds \) | \(=\) | \(\ds \cos a \cos b + \sin a \sin b\) | Sine Function is Odd |
$\blacksquare$
Proof 2
Consider two radii $OP$ and $OQ$ of a unit circle whose center is at the origin of a Cartesian plane.
Let:
| \(\ds \angle xOP\) | \(=\) | \(\ds B\) | ||||||||||||
| \(\ds \angle xOQ\) | \(=\) | \(\ds A\) |
Then the coordinates of $P$ and $Q$ are given by:
| \(\ds P\) | \(=\) | \(\ds \tuple {\cos B, \sin B}\) | ||||||||||||
| \(\ds Q\) | \(=\) | \(\ds \tuple {\cos A, \sin A}\) |
Hence:
| \(\ds PQ^2\) | \(=\) | \(\ds \paren {\cos B - \cos A}^2 + \paren {\sin B - \sin A}^2\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \cos^2 B - 2 \cos A \cos B + \cos^2 A + \sin^2 B - 2 \sin A \sin B + \sin^2 A\) | multiplying out | |||||||||||
| \(\ds \) | \(=\) | \(\ds 2 - 2 \cos A \cos B - 2 \sin A \sin B\) | Sum of Squares of Sine and Cosine | |||||||||||
| \(\ds \) | \(=\) | \(\ds 1 + 1 - 2 \map \cos {A - B}\) | Law of Cosines, as $\angle POQ = A - B$ | |||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \map \cos {A - B}\) | \(=\) | \(\ds \cos A \cos B + \sin A \sin B\) | simplifying |
$\blacksquare$
Also see
Historical Note
The Cosine of Sum formula and its were proved by François Viète in about $1579$.
Sources
- 1953: L. Harwood Clarke: A Note Book in Pure Mathematics ... (previous) ... (next): $\text V$. Trigonometry: Formulae $(9)$
- 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: $4$
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): addition formulae
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): addition formulae
- 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): compound angle formulae (in trigonometry)
- 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): Appendix $12$: Trigonometric formulae: Addition formulae
- 2021: Richard Earl and James Nicholson: The Concise Oxford Dictionary of Mathematics (6th ed.) ... (previous) ... (next): Appendix $14$: Trigonometric formulae: Addition formulae