Cosinus versus

The versine or versed sine is a trigonometric function already appearing in some of the earliest trigonometric tables. The versine of an angle equals 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.

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

Expressed in terms of the meanwhile more commonly used "vertical" sines (sinus rectus) and cosines (cosinus rectus) functions, the versine is equal to 1 − cos(θ), or 2 sin²(θ/2).

There are several related functions corresponding to the versine:

In full analogy to the above-mentioned four functions another set of four "half-value" functions exists as well:

The ordinary sine function (see note on etymology) was sometimes historically called the sinus rectus ("vertical sine"), to contrast it with the versed sine (sinus versus). The meaning of these terms is apparent if one looks at the functions in the original context for their definition, a unit circle:

For a vertical chord AB of the unit circle, the sine of the angle θ (representing half of the subtended angle Δ) is the distance AC (half of the chord). On the other hand, the versed sine of θ is the distance CD from the center of the chord to the center of the arc. Thus, the sum of cos(θ) (equal to the length of line OC) and versin(θ) (equal to the length of line CD) is the radius OD (with length 1). Illustrated this way, the sine is vertical (rectus, literally "straight") while the versine is horizontal (versus, literally "turned against, out-of-place"); both are distances from C to the circle.

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