# Prosthaphaeresis Formulas/Sine plus Sine

## Theorem

- $\sin \alpha + \sin \beta = 2 \sin \left({\dfrac {\alpha + \beta} 2}\right) \cos \left({\dfrac {\alpha - \beta} 2}\right)$

where $\sin$ denotes sine and $\cos$ denotes cosine.

## Proof

\(\displaystyle \) | \(\) | \(\displaystyle 2 \sin \left({\frac {\alpha + \beta} 2}\right) \cos \left({\frac {\alpha - \beta} 2}\right)\) | |||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle 2 \frac {\sin \left({\dfrac {\alpha + \beta} 2 + \dfrac {\alpha - \beta} 2}\right) + \sin \left({\dfrac {\alpha + \beta} 2 - \dfrac {\alpha - \beta} 2}\right)} 2\) | Simpson's Formula for Sine by Cosine | ||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \sin \frac {2 \alpha} 2 + \sin \frac {2 \beta} 2\) | |||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \sin \alpha + \sin \beta\) |

$\blacksquare$

## Also reported as

This result is also sometimes reported as:

- $\dfrac {\sin \alpha + \sin \beta} 2 = \sin \left({\dfrac {\alpha + \beta} 2}\right) \cos \left({\dfrac {\alpha - \beta} 2}\right)$

## 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

- 1968: Murray R. Spiegel:
*Mathematical Handbook of Formulas and Tables*... (previous) ... (next): $\S 5$: Trigonometric Functions: $5.61$ - 2008: Ian Stewart:
*Taming the Infinite*... (previous) ... (next): Chapter $5$: Eternal Triangles