Sine of Difference
Corollary to Sine of Sum
- $\map \sin {a - b} = \sin a \cos b - \cos a \sin b$
where $\sin$ denotes the sine and $\cos$ denotes the cosine.
Proof 1
| \(\ds \map \sin {a - b}\) | \(=\) | \(\ds \sin a \map \cos {-b} + \cos a \map \sin {-b}\) | Sine of Sum | |||||||||||
| \(\ds \) | \(=\) | \(\ds \sin a \cos b + \cos a \, \map \sin {-b}\) | Cosine Function is Even | |||||||||||
| \(\ds \) | \(=\) | \(\ds \sin a \cos b - \cos a \sin b\) | Sine Function is Odd |
$\blacksquare$
Proof 2
| \(\ds \map \cos {90 \degrees + a - b}\) | \(=\) | \(\ds \map \cos {90 \degrees + a} \cos b + \map \sin {90 \degrees + a} \sin b\) | Cosine of Difference | |||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \map \sin {a - b}\) | \(=\) | \(\ds \sin a \cos b - \cos a \sin b\) | Cosine of Angle plus Right Angle, Sine of Angle plus Right Angle |
$\blacksquare$
Also see
Historical Note
The Sine 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 $(10)$
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 5$: Trigonometric Functions: $5.34$
- 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