The versine or versed sine is a
trigonometric function
In mathematics, the trigonometric functions (also called circular functions, angle functions or goniometric functions) are real functions which relate an angle of a right-angled triangle to ratios of two side lengths. They are widely used in a ...
found in some of the earliest (
Sanskrit
Sanskrit (; attributively , ; nominally , , ) is a classical language belonging to the Indo-Aryan branch of the Indo-European languages. It arose in South Asia after its predecessor languages had diffused there from the northwest in the late ...
''Aryabhatia'',
[The Āryabhaṭīya by Āryabhaṭa]
Section I)
trigonometric table
In mathematics, tables of trigonometric functions are useful in a number of areas. Before the existence of pocket calculators, trigonometric tables were essential for navigation, science and engineering. The calculation of mathematical tables w ...
s. 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 The haversine formula determines the great-circle distance between two points on a sphere given their longitudes and latitudes. Important in navigation, it is a special case of a more general formula in spherical trigonometry, the law of haversines, ...
of navigation.
Overview
The versine
or versed sine
is a
trigonometric function
In mathematics, the trigonometric functions (also called circular functions, angle functions or goniometric functions) are real functions which relate an angle of a right-angled triangle to ratios of two side lengths. They are widely used in a ...
already appearing in some of the earliest trigonometric tables. It is symbolized in formulas using the abbreviations , ,
,
or .
In
Latin
Latin (, or , ) is a classical language belonging to the Italic branch of the Indo-European languages. Latin was originally a dialect spoken in the lower Tiber area (then known as Latium) around present-day Rome, but through the power of the ...
, it is known as the ''sinus versus'' (flipped sine), ''versinus'', ''versus'', or ''sagitta'' (arrow).
Expressed in terms of common
trigonometric functions
In mathematics, the trigonometric functions (also called circular functions, angle functions or goniometric functions) are real functions which relate an angle of a right-angled triangle to ratios of two side lengths. They are widely used in al ...
sine, cosine, and tangent, the versine is equal to
There are several related functions corresponding to the versine:
* The versed cosine,
or vercosine, abbreviated , , or .
* The coversed sine or coversine
(in Latin, ''cosinus versus'' or ''coversinus''), abbreviated ,
,
, or
* The coversed cosine
or covercosine, abbreviated , , or
In full analogy to the above-mentioned four functions another set of four "half-value" functions exists as well:
* The haversed sine
or haversine (Latin ''semiversus''),
abbreviated , , , , ,
,
, or ,
most famous from the
haversine formula The haversine formula determines the great-circle distance between two points on a sphere given their longitudes and latitudes. Important in navigation, it is a special case of a more general formula in spherical trigonometry, the law of haversines, ...
used historically in
navigation
Navigation is a field of study that focuses on the process of monitoring and controlling the movement of a craft or vehicle from one place to another.Bowditch, 2003:799. The field of navigation includes four general categories: land navigation, ...
* The haversed cosine
or havercosine, abbreviated , , or
* The hacoversed sine, hacoversine,
or cohaversine, abbreviated , , ,
or
* The hacoversed cosine,
hacovercosine, or cohavercosine, abbreviated , or
History and applications
Versine and coversine
![historical_trigonometric_functions_graph](https://upload.wikimedia.org/wikipedia/commons/4/46/Historical_trigonometric_functions_graph.svg)
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'').
The meaning of these terms is apparent if one looks at the functions in the original context for their definition, a
unit circle
In mathematics, a unit circle is a circle of unit radius—that is, a radius of 1. Frequently, especially in trigonometry, the unit circle is the circle of radius 1 centered at the origin (0, 0) in the Cartesian coordinate system in the Eucli ...
:
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.
This figure also illustrates the reason why the versine was sometimes called the ''sagitta'', Latin for
arrow,
from the Arabic usage ''sahem''
of the same meaning. This itself comes from the Indian word 'sara' (arrow) that was commonly used to refer to "
utkrama-jya". If the arc ''ADB'' of the double-angle ''Δ'' = 2''θ'' is viewed as a "
bow" 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
abscissa
In common usage, the abscissa refers to the (''x'') coordinate and the ordinate refers to the (''y'') coordinate of a standard two-dimensional graph.
The distance of a point from the y-axis, scaled with the x-axis, is called abscissa or x coo ...
(the horizontal axis of a graph).
In 1821,
Cauchy
Baron Augustin-Louis Cauchy (, ; ; 21 August 178923 May 1857) was a French mathematician, engineer, and physicist who made pioneering contributions to several branches of mathematics, including mathematical analysis and continuum mechanics. He w ...
used the terms ''sinus versus'' (''siv'') for the versine and ''cosinus versus'' (''cosiv'') for the coversine.
Historically, the versed sine was considered one of the most important trigonometric functions.
As ''θ'' goes to zero, versin(''θ'') is the difference between two nearly equal quantities, so a user of a
trigonometric table
In mathematics, tables of trigonometric functions are useful in a number of areas. Before the existence of pocket calculators, trigonometric tables were essential for navigation, science and engineering. The calculation of mathematical tables w ...
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.
Even with a calculator or computer,
round-off errors make it advisable to use the sin
2 formula for small ''θ''.
Another historical advantage of the versine is that it is always non-negative, so its
logarithm
In mathematics, the logarithm is the inverse function to exponentiation. That means the logarithm of a number to the base is the exponent to which must be raised, to produce . For example, since , the ''logarithm base'' 10 of ...
is defined everywhere except for the single angle (''θ'' = 0, 2, …) where it is zero—thus, one could use
logarithmic tables for multiplications in formulas involving versines.
In fact, the earliest surviving table of sine (half-
chord) values (as opposed to the
chords tabulated by Ptolemy and other Greek authors), calculated from the
Surya Siddhantha of India dated back to the 3rd century BC, was a table of values for the sine and versed sine (in 3.75° increments from 0 to 90°).
The versine appears as an intermediate step in the application of the
half-angle formula
In trigonometry, trigonometric identities are equalities that involve trigonometric functions and are true for every value of the occurring variables for which both sides of the equality are defined. Geometrically, these are identities involvin ...
sin
2 = versin(''θ''), derived by
Ptolemy
Claudius Ptolemy (; grc-gre, Πτολεμαῖος, ; la, Claudius Ptolemaeus; AD) was a mathematician, astronomer, astrologer, geographer, and music theorist, who wrote about a dozen scientific treatises, three of which were of importance ...
, that was used to construct such tables.
Haversine
The haversine, in particular, was important in
navigation
Navigation is a field of study that focuses on the process of monitoring and controlling the movement of a craft or vehicle from one place to another.Bowditch, 2003:799. The field of navigation includes four general categories: land navigation, ...
because it appears in the
haversine formula The haversine formula determines the great-circle distance between two points on a sphere given their longitudes and latitudes. Important in navigation, it is a special case of a more general formula in spherical trigonometry, the law of haversines, ...
, which is used to reasonably accurately compute distances on an astronomic
spheroid
A spheroid, also known as an ellipsoid of revolution or rotational ellipsoid, is a quadric surface obtained by rotating an ellipse about one of its principal axes; in other words, an ellipsoid with two equal semi-diameters. A spheroid has ...
(see issues with the
earth's radius vs. sphere) given angular positions (e.g.,
longitude
Longitude (, ) is a geographic coordinate that specifies the east– west position of a point on the surface of the Earth, or another celestial body. It is an angular measurement, usually expressed in degrees and denoted by the Greek lette ...
and
latitude
In geography, latitude is a coordinate that specifies the north– south position of a point on the surface of the Earth or another celestial body. Latitude is given as an angle that ranges from –90° at the south pole to 90° at the north pol ...
). One could also use sin
2 directly, but having a table of the haversine removed the need to compute squares and square roots.
An early utilization by
José de Mendoza y Ríos of what later would be called haversines is documented in 1801.
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".
In 1835, the term ''
haversine
The versine or versed sine is a trigonometric function found in some of the earliest (Sanskrit ''Aryabhatia'',[base-10 logarithmically as ''log. haversine'' or ''log. havers.'') was coined] by James Inman
James Inman (1776–1859), an English mathematician and astronomer, was professor of mathematics at the Royal Naval College, Portsmouth, and author of ''Inman's Nautical Tables''.
Early years
Inman was born at Tod Hole in Garsdale, then in the ...
in the third edition of his work ''Navigation and Nautical Astronomy: For the Use of British Seamen'' to simplify the calculation of distances between two points on the surface of the earth using spherical trigonometry
Spherical trigonometry is the branch of spherical geometry that deals with the metrical relationships between the sides and angles of spherical triangles, traditionally expressed using trigonometric functions. On the sphere, geodesics are grea ...
for applications in navigation. Inman also used the terms ''nat. versine'' and ''nat. vers.'' for versines.
Other high-regarded tables of haversines were those of Richard Farley in 1856 and John Caulfield Hannyngton in 1876.
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 logarithm
In mathematics, addition and subtraction logarithms or Gaussian logarithms can be utilized to find the logarithms of the Addition, sum and Subtraction, difference of a pair of values whose logarithms are known, without knowing the values themselves ...
s since 1995 or in a more compact method for sight reduction In astronavigation, sight reduction is the process of deriving from a sight, (in celestial navigation usually obtained using a sextant), the information needed for establishing a line of position, generally by intercept method.
Sight is defined ...
since 2014.
Modern uses
Whilst the usage of the versine, coversine and haversine as well as their inverse function
In mathematics, the inverse function of a function (also called the inverse of ) is a function that undoes the operation of . The inverse of exists if and only if is bijective, and if it exists, is denoted by f^ .
For a function f\colon X ...
s can be traced back centuries, the names for the other five cofunction
In mathematics, a function (mathematics), function ''f'' is cofunction of a function ''g'' if ''f''(''A'') = ''g''(''B'') whenever ''A'' and ''B'' are complementary angles. This definition typically applies to trigonometric functions. The prefix ...
s appear to be of much younger origin.
One period (0 < ''θ'' < ) of a versine or, more commonly, a haversine (or havercosine) waveform is also commonly used in signal processing
Signal processing is an electrical engineering subfield that focuses on analyzing, modifying and synthesizing ''signals'', such as sound, images, and scientific measurements. Signal processing techniques are used to optimize transmissions, ...
and control theory
Control theory is a field of mathematics that deals with the control of dynamical systems in engineered processes and machines. The objective is to develop a model or algorithm governing the application of system inputs to drive the system to a ...
as the shape of a pulse
In medicine, a pulse represents the tactile arterial palpation of the cardiac cycle (heartbeat) by trained fingertips. The pulse may be palpated in any place that allows an artery to be compressed near the surface of the body, such as at the n ...
or a window function
In signal processing and statistics, a window function (also known as an apodization function or tapering function) is a mathematical function that is zero-valued outside of some chosen interval, normally symmetric around the middle of the int ...
(including Hann, Hann–Poisson and Tukey window
In discrete-time signal processing, windowing is a preliminary signal shaping technique, usually applied to improve the appearance and usefulness of a subsequent Discrete Fourier Transform. Several '' window functions'' can be defined, based on ...
s), because it smoothly (continuous
Continuity or continuous may refer to:
Mathematics
* Continuity (mathematics), the opposing concept to discreteness; common examples include
** Continuous probability distribution or random variable in probability and statistics
** Continuous ...
in value and slope
In mathematics, the slope or gradient of a line is a number that describes both the ''direction'' and the ''steepness'' of the line. Slope is often denoted by the letter ''m''; there is no clear answer to the question why the letter ''m'' is use ...
) "turns on" from zero
0 (zero) is a number representing an empty quantity. In place-value notation such as the Hindu–Arabic numeral system, 0 also serves as a placeholder numerical digit, which works by multiplying digits to the left of 0 by the radix, usual ...
to one (for haversine) and back to zero. In these applications, it is named Hann function
The Hann function is named after the Austrian meteorologist Julius von Hann. It is a window function used to perform Hann smoothing. The function, with length L and amplitude 1/L, is given by:
:
w_0(x) \triangleq \left\.
For digital s ...
or raised-cosine filter The raised-cosine filter is a filter frequently used for pulse-shaping in digital modulation due to its ability to minimise intersymbol interference (ISI). Its name stems from the fact that the non-zero portion of the frequency spectrum of its simp ...
. Likewise, the havercosine is used in raised-cosine distribution
In probability theory and statistics, the raised cosine distribution is a continuous probability distribution support (mathematics), supported on the interval mu-s,\mu+s The probability density function (PDF) is
:f(x;\mu,s)=\frac
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s_in_[
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