Versine

Versine (vers) used to be a very common Trigonometry function, but it is now very unpopular.

$vers(\theta )=1-\cos(\theta )$

Properties

The derivative of $vers(\theta )$is $\sin(\theta)$ while the antiderivative is $\theta -\sin(\theta )+C$.

The square of versine:

$vers^{2}(\theta )=2\sin ^{2}({\frac {\theta }{2}})$

History

The versine was historically used in navigation and surveying, particularly in calculating distances on the Earth's surface and assessing track alignment. In navigation, it was part of the haversine formula for distance calculations on a sphere. In surveying and rail transport, versines were used to measure the curvature and straightness of tracks.