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:


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

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