Versine
[18] Expressed in terms of common trigonometric functions sine, cosine, and tangent, the versine is equal to
[37] 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).
[18][36] If the arc ADB of the double-angle Δ = 2θ is viewed as a "bow" and the chord AB as its "string", then the versine CD is clearly the "arrow shaft".
[12][37][38] As θ goes to zero, versin(θ) is the difference between two nearly equal quantities, so a user of a trigonometric table for the cosine alone would need a very high accuracy to obtain the versine in order to avoid catastrophic cancellation, making separate tables for the latter convenient.
[12] Even with a calculator or computer, round-off errors make it advisable to use the sin2 formula for small θ.
[37] The versine appears as an intermediate step in the application of the half-angle formula sin2(θ/2) = 1/2versin(θ), derived by Ptolemy, that was used to construct such tables.
[12] An early utilization by José de Mendoza y Ríos of what later would be called haversines is documented in 1801.
[14][39] The first known English equivalent to a table of haversines was published by James Andrew in 1805, under the name "Squares of Natural Semi-Chords".
[40][46] The haversine continues to be used in navigation and has found new applications in recent decades, as in Bruce D. Stark's method for clearing lunar distances utilizing Gaussian logarithms since 1995[47][48] or in a more compact method for sight reduction since 2014.
[32] While the usage of the versine, coversine and haversine as well as their inverse functions can be traced back centuries, the names for the other five cofunctions appear to be of much younger origin.
One period (0 < θ < 2π) of a versine or, more commonly, a haversine (or havercosine) waveform is also commonly used in signal processing and control theory as the shape of a pulse or a window function (including Hann, Hann–Poisson and Tukey windows), because it smoothly (continuous in value and slope) "turns on" from zero to one (for haversine) and back to zero.
In the form of sin2(θ) the haversine of the double-angle Δ describes the relation between spreads and angles in rational trigonometry, a proposed reformulation of metrical planar and solid geometries by Norman John Wildberger since 2005.
Inverse functions like arcversine[34] (arcversin, arcvers,[8][34] avers,[51][52] aver), arcvercosine (arcvercosin, arcvercos, avercos, avcs), arccoversine[34] (arccoversin, arccovers,[8][34] acovers,[51][52] acvs), arccovercosine (arccovercosin, arccovercos, acovercos, acvc), archaversine (archaversin, archav,[34] haversin−1,[53] invhav,[34][54][55][56] ahav,[34][51][52] ahvs, ahv, hav−1[57][58]), archavercosine (archavercosin, archavercos, ahvc), archacoversine (archacoversin, ahcv) or archacovercosine (archacovercosin, archacovercos, ahcc) exist as well: These functions can be extended into the complex plane.
[50][20][27] Maclaurin series:[27] When the versine v is small in comparison to the radius r, it may be approximated from the half-chord length L (the distance AC shown above) by the formula[59]
The term versine is also sometimes used to describe deviations from straightness in an arbitrary planar curve, of which the above circle is a special case.
The term sagitta (often abbreviated sag) is used similarly in optics, for describing the surfaces of lenses and mirrors.
