| • Science | • People | • Locations | • Timeline |
There are also three corresponding functions: the coversed sine (the versed sine of the complementary angle π/2 − θ, or coversine), the haversed sine or haversine (half the versed sine), and the hacoversed sine (half the coversed sine, also called the hacoversine, the cohaversine, and the havercosine):
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, floating point 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 zero—thus, 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 (see references).
In fact, the earliest surviving trigonometric table, from the 4th– 5th century Siddhantas from India, was a table of values for the sine and versed sine only (in 3.75-degree increments from 0 to 90 degrees). This is, perhaps, even less surprising considering that 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.
This figure also illustrates the reason why the versine was sometimes called the sagitta, Latin for arrowThis is an article about the projectile; see Arrow (disambiguation) for other meanings. An arrow is a pointed projectile that is shot with a bow. It predates history and is common to most cultures. An arrow consists of a long and thin shaft made formerly, from the Arabic usage sahem of the same meaning. If the arc ADB is viewed as a " bowA bow is a weapon that shoots arrows powered by the elasticity of the bow and/or the string. It is useful for hunting and war. The technique of using a bow is called archery. A large number of different bow designs have been used in different cultures and" and the chord AB as its "string", then the versine CD is clearly the "arrow shaft".
In further keeping with the interpretation of the sine as "vertical" and the versed sine as "horizontal", sagitta is also an obsolete synonym for the abscissaAbscissa means the x coordinate on an (x, y) graph; the input of a mathematical function against which the output is plotted. y is the "ordinate". See also Cartesian coordinate Mathematics References Calculus. (the horizontal axis of a graph).
One period (θ = 0..2π) of a versine or, more commonly, a haversine waveform is also commonly used in signal processingSignal processing is the processing, amplification and interpretation of signals. Signals may come from various sources. There are various sorts of signal processing, depending on the nature of the signal, as in the following examples. Digital signal proc and control theoryThis article is about an engineering theory called control theory. There is also a sociological theory of deviant behavior that is called control theory. In engineering and mathematics, control theory deals with the behaviour of dynamical systems over tim as the shape of a pulse or a window function, because it smoothly ( continuous in value and slope) "turns on" from zero to one (for haversine) and back to zero.