Versine

The versine or versed sine is a trigonometric function found in some of the earliest (Sanskrit Aryabhatiya,

Section I) trigonometric tables. The versine of an angle is 1 minus its cosine.

There are several related functions, most notably the coversine and haversine. The latter, half a versine, is of particular importance in the haversine formula of navigation.

Overview

The versine or versed sine is a trigonometric function already appearing in some of the earliest trigonometric tables. It is symbolized in formulas using the abbreviations versin, sinver, vers, or siv. In Latin, it is known as the sinus versus (flipped sine), versinus, versus, or sagitta (arrow).

Expressed in terms of common trigonometric functions sine, cosine, and tangent, the versine is equal to

versin

⁡

θ

=

1

−

cos

⁡

θ

=

2

sin

2

⁡

θ

2

=

sin

⁡

θ

tan

⁡

θ

2

{\displaystyle \operatorname {versin} \theta =1-\cos \theta =2\sin ^{2}{\frac {\theta }{2}}=\sin \theta \,\tan {\frac {\theta }{2}}}

There are several related functions corresponding to the versine:

The versed cosine, or vercosine, abbreviated vercosin, vercos, or vcs

The coversed sine or coversine (in Latin, cosinus versus or coversinus), abbreviated coversin, covers, cosiv, or cvs

Special tables were also made of half of the versed sine, because of its particular use in the haversine formula used historically in navigation.

The haversed sine or haversine (Latin: semiversus), abbreviated haversin, semiversin, semiversinus, havers, hav, hvs, sem, or hv. It is defined as

hav

⁡

θ

=

sin

2

⁡

(

θ

2

)

=

1

−

cos

⁡

θ

2

{\displaystyle \operatorname {hav} \theta =\sin ^{2}\left({\frac {\theta }{2}}\right)={\frac {1-\cos \theta }{2}}}

History and applications

Versine and coversine

The ordinary sine function (see note on etymology) was sometimes historically called the sinus rectus ("straight sine"), to contrast it with the versed sine (sinus versus).