# Definition:Haversine

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## Definition

The **haversine** of an angle is defined as:

\(\displaystyle \hav \theta\) | \(=\) | \(\displaystyle \dfrac 1 2 \paren {1 - \cos \theta}\) | |||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \dfrac 1 2 \vers \theta\) | where $\vers \theta$ denotes the versed sine of $\theta$ | ||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \sin^2 \dfrac {\theta} 2\) | Double Angle Formula for Cosine: Corollary 2 |

## Also see

- Results about
**haversines**can be found here.

## Historical Note

In the days before digital computers, log tables and slide rules were de rigueur in all fields of mathematics where calculation was involved.

Calculation of distances and directions on the surface of Earth using the Spherical Law of Cosines and its derived formulas is computation-intensive.

The Spherical Law of Haversines is easier to use, assuming one has a table of haversines and logarithms of haversines.

Hence use of the **haversine** was commonplace in nautical context.

## Linguistic Note

The word **haversine** derives from **half versed sine**.

## Sources

- 1976: W.M. Smart:
*Textbook on Spherical Astronomy*(6th ed.) ... (previous) ... (next): Chapter $\text I$. Spherical Trigonometry: $13$.*The haversine formula.* - 1989: Ephraim J. Borowski and Jonathan M. Borwein:
*Dictionary of Mathematics*... (previous) ... (next): Entry:**haversine**