versine and other functions
Versine
The versine or versed sine, versin (θ), is a trigonometric function equal to 1 − cos (θ) and 2sin2(½θ).
The versine or versed sine, versin (θ), is a trigonometric functionequal to 1 − cos (θ) and 2sin2(½θ).
There are several related functions, most notably the haversine, half the versine, known in the haversine formula of navigation. It is also written as vers(θ) or ver(θ). In Latin, it is known as the sinus versus (flipped sine) or the sagitta (arrow).
History and applications
Historically, the versed sine was considered one of the most important trigonometric functions, but it has fallen from popularity in modern times due to the availability of computers and scientific calculators.
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, making separate tables for the latter convenient.
Even with a computer, round-off errors make it advisable to use the sin2 formula for small θ. Another historical advantage of the versine is that it is always non-negative, so its logarithm is defined everywhere except for the single angle (θ = 0, 2π,...) where it is zerothus, one could use logarithmic tables for multiplications in formulas involving versines.
The haversine, in particular, was important in navigation because it appears in the haversine formula, which is used to accurately compute distances on a sphere given angular positions (e.g., longitude and latitude).
One could also use sin2(θ/2) directly, but having a table of the haversine removed the need to compute squares and square roots. The term haversine was, apparently, coined in a navigation text for just such an application.
In fact, the earliest surviving table of sine (half-chord) values (as opposed to the chords tabulated by Ptolemy and other Greek authors), from the fourth–fifth century Siddhantas from India, was a table of values for the sine and versed sine only (in 3.75° increments from 0 to 90°). The versine appears as an intermediate step in the application of the half-angle formula sin2(θ/2) = versin(θ)/2, derived by Ptolemy, that was used to construct such tables.
