Prosthaphaeresis Formulas/Sine plus Sine
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Theorem
- $\sin \alpha + \sin \beta = 2 \map \sin {\dfrac {\alpha + \beta} 2} \map \cos {\dfrac {\alpha - \beta} 2}$
where $\sin$ denotes sine and $\cos$ denotes cosine.
Proof 1
\(\text {(1)}: \quad\) | \(\ds \map \sin {A + B}\) | \(=\) | \(\ds \sin A \cos B + \cos A \sin B\) | Sine of Sum | ||||||||||
\(\text {(2)}: \quad\) | \(\ds \map \sin {A - B}\) | \(=\) | \(\ds \sin A \cos B - \cos A \sin B\) | Sine of Difference | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds \map \sin {A + B} + \map \sin {A - B}\) | \(=\) | \(\ds 2 \sin A \cos B\) | adding $(1)$ and $(2)$ | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds \sin \alpha + \sin \beta\) | \(=\) | \(\ds 2 \map \sin {\dfrac {\alpha + \beta} 2} \map \cos {\dfrac {\alpha - \beta} 2}\) | setting $A + B = \alpha$ and $A - B = \beta$ |
$\blacksquare$
Proof 2
\(\ds \) | \(\) | \(\ds 2 \map \sin {\dfrac {\alpha + \beta} 2} \map \cos {\dfrac {\alpha - \beta} 2}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 2 \frac {\map \sin {\dfrac {\alpha + \beta} 2 + \dfrac {\alpha - \beta} 2} + \map \sin {\dfrac {\alpha + \beta} 2 - \dfrac {\alpha - \beta} 2} } 2\) | Werner Formula for Sine by Cosine | |||||||||||
\(\ds \) | \(=\) | \(\ds \sin \frac {2 \alpha} 2 + \sin \frac {2 \beta} 2\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \sin \alpha + \sin \beta\) |
$\blacksquare$
Also reported as
This result is also sometimes reported as:
- $\dfrac {\sin \alpha + \sin \beta} 2 = \map \sin {\dfrac {\alpha + \beta} 2} \map \cos {\dfrac {\alpha - \beta} 2}$
Also known as
The Prosthaphaeresis Formulas are also known as:
- the Factor Formulas (or Factor Formulae)
- Simpson's Formulas (or Simpson's Formulae), although this is usually used for similar results.
Examples
$\sin 40 \degrees$ plus $\sin 60 \degrees$
- $\sin 40 \degrees + \sin 60 \degrees = 2 \sin 50 \degrees \cos 10 \degrees$
$\sin 2 A$ plus $\sin 6 A$
- $\sin 2 A + \sin 6 A = 2 \sin 4 A \cos 2 A$
Solution to $\sin 2 x + \sin 5 x = \sin 4 x$
The equation
- $\sin 3 x + \sin 5 x = \sin 4 x$
has the general solution:
- $\set {\dfrac {n \pi} 4 : n \in \Z} \cup \set {2 n \pi \pm \dfrac \pi 3: n \in \Z}$
Also see
Linguistic Note
The word prosthaphaeresis or prosthapheiresis is a neologism coined some time in the $16$th century from the two Greek words:
- prosthesis, meaning addition
- aphaeresis or apheiresis, meaning subtraction.
With the advent of machines to aid the process of arithmetic, this word now has only historical significance.
Ian Stewart, in his Taming the Infinite from $2008$, accurately and somewhat diplomatically describes the word as "ungainly".
Sources
- 1953: L. Harwood Clarke: A Note Book in Pure Mathematics ... (previous) ... (next): $\text V$. Trigonometry: Formulae $(28)$
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 5$: Trigonometric Functions: $5.61$
- 1969: J.C. Anderson, D.M. Hum, B.G. Neal and J.H. Whitelaw: Data and Formulae for Engineering Students (2nd ed.) ... (previous) ... (next): $4.$ Mathematics: $4.3$ Trigonometric identities and hyperbolic functions
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): factor formulae
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): factor formulae
- 2008: Ian Stewart: Taming the Infinite ... (previous) ... (next): Chapter $5$: Eternal Triangles: Napierian logarithms
- 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: Sums and differences
- 2021: Richard Earl and James Nicholson: The Concise Oxford Dictionary of Mathematics (6th ed.) ... (previous) ... (next): Appendix $14$: Trigonometric formulae: Factor formulae