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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 all ...
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 all ...
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 all ...
sine, cosine, and tangent, the versine is equal to \operatorname\theta = 1 - \cos \theta = 2\sin^\frac\theta2 = \sin\theta\,\tan\frac\theta2 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

The ordinary ''
sine In mathematics, sine and cosine are trigonometric functions of an angle. The sine and cosine of an acute angle are defined in the context of a right triangle: for the specified angle, its sine is the ratio of the length of the side that is oppo ...
'' 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 Eucl ...
: 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 An arrow is a fin-stabilized projectile launched by a bow. A typical arrow usually consists of a long, stiff, straight shaft with a weighty (and usually sharp and pointed) arrowhead attached to the front end, multiple fin-like stabilizers c ...
, 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 In numerical analysis, catastrophic cancellation is the phenomenon that subtracting good approximations to two nearby numbers may yield a very bad approximation to the difference of the original numbers. For example, if there are two studs, one L_ ...
, making separate tables for the latter convenient. Even with a calculator or computer,
round-off error A roundoff error, also called rounding error, is the difference between the result produced by a given algorithm using exact arithmetic and the result produced by the same algorithm using finite-precision, rounded arithmetic. Rounding errors are d ...
s 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 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 o ...
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 ...
sin2 = 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 importanc ...
, 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 cir ...
(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 letter l ...
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 sin2 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 José is a predominantly Spanish and Portuguese form of the given name Joseph. While spelled alike, this name is pronounced differently in each language: Spanish ; Portuguese (or ). In French, the name ''José'', pronounced , is an old vernacul ...
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 logarithm In mathematics, the common logarithm is the logarithm with base 10. It is also known as the decadic logarithm and as the decimal logarithm, named after its base, or Briggsian logarithm, after Henry Briggs, an English mathematician who pioneered ...
ically 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 gr ...
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 as ...
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\t ...
s can be traced back centuries, the names for the other five
cofunction In mathematics, a 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 "co-" can be found alrea ...
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 audio signal processing, sound, image processing, images, and scientific measurements. Signal processing techniq ...
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 nec ...
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 inte ...
(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 Positional notation (or place-value notation, or positional numeral system) usually denotes the extension to any base of the Hindu–Arabic numeral system (or ...
to
one 1 (one, unit, unity) is a number representing a single or the only entity. 1 is also a numerical digit and represents a single unit of counting or measurement. For example, a line segment of ''unit length'' is a line segment of length 1. I ...
(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 sign ...
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 sim ...
. 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 \left +\cos\_...
s_in_s_in_probability_theory_and_statistics">probability_theory.html"_;"title="+\cos\_...
s_in_probability_theory">+\cos\_...
s_in_probability_theory_and_statistics. In_the_form_of_sin2(''θ'')_the_haversine_of_the_double-angle_''Δ''_describes_the_relation_between_spread_(rational_trigonometry).html" "title="probability_theory_and_statistics.html" ;"title="probability_theory.html" ;"title="+\cos\ ...
s in probability theory">+\cos\ ...
s in probability theory and statistics">probability_theory.html" ;"title="+\cos\ ...
s in probability theory">+\cos\ ...
s in probability theory and statistics. In the form of sin2(''θ'') the haversine of the double-angle ''Δ'' describes the relation between spread (rational trigonometry)">spread Spread may refer to: Places * Spread, West Virginia Arts, entertainment, and media * ''Spread'' (film), a 2009 film. * ''$pread'', a quarterly magazine by and for sex workers * "Spread", a song by OutKast from their 2003 album ''Speakerboxxx/T ...
s and angles in rational trigonometry, a proposed reformulation of metric space, metrical plane geometry, planar and solid geometry, solid geometries by Norman John Wildberger since 2005.


Mathematical identities


Definitions


Circular rotations

The functions are circular rotations of each other. :\begin \mathrm(\theta) &= \mathrm\left(\theta + \frac\right) = \mathrm\left(\theta + \pi\right) = \mathrm\left(\theta + \frac\right) \\ \mathrm(\theta) &= \mathrm\left(\theta + \frac\right) = \mathrm\left(\theta + \pi\right) = \mathrm\left(\theta + \frac\right) \end


Derivatives and integrals


Inverse functions

Inverse functions like arcversine (arcversin, arcvers, avers, aver), arcvercosine (arcvercosin, arcvercos, avercos, avcs), arccoversine (arccoversin, arccovers, acovers, acvs), arccovercosine (arccovercosin, arccovercos, acovercos, acvc), archaversine (archaversin, archav, haversin−1, invhav, ahav, ahvs, ahv, hav−1), archavercosine (archavercosin, archavercos, ahvc), archacoversine (archacoversin, ahcv) or archacovercosine (archacovercosin, archacovercos, ahcc) exist as well:


Other properties

These functions can be extended into the
complex plane In mathematics, the complex plane is the plane formed by the complex numbers, with a Cartesian coordinate system such that the -axis, called the real axis, is formed by the real numbers, and the -axis, called the imaginary axis, is formed by the ...
.
Maclaurin series Maclaurin or MacLaurin is a surname. Notable people with the surname include: * Colin Maclaurin (1698–1746), Scottish mathematician * Normand MacLaurin (1835–1914), Australian politician and university administrator * Henry Normand MacLaurin ( ...
: : \begin \operatorname(z) &= \sum_^\infty \frac \\ \operatorname(z) &= \sum_^\infty \frac \end : \lim_ \frac = 0 : \begin \frac - \frac &= \frac \\ pt operatorname(\theta) + \operatorname(\theta), operatorname(\theta) + \operatorname(\theta) &= \sin(\theta) \cos(\theta) \end


Approximations

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 v \approx \frac. Alternatively, if the versine is small and the versine, radius, and half-chord length are known, they may be used to estimate the arc length ''s'' (''AD'' in the figure above) by the formula s\approx L+\frac This formula was known to the Chinese mathematician
Shen Kuo Shen Kuo (; 1031–1095) or Shen Gua, courtesy name Cunzhong (存中) and pseudonym Mengqi (now usually given as Mengxi) Weng (夢溪ç¿),Yao (2003), 544. was a Chinese polymathic scientist and statesman of the Song dynasty (960–1279). Shen wa ...
, and a more accurate formula also involving the sagitta was developed two centuries later by
Guo Shoujing Guo Shoujing (, 1231–1316), courtesy name Ruosi (), was a Chinese astronomer, hydraulic engineer, mathematician, and politician of the Yuan dynasty. The later Johann Adam Schall von Bell (1591–1666) was so impressed with the preserved astron ...
. A more accurate approximation used in engineering is v\approx \frac


Arbitrary curves and chords

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. Given a chord between two points in a curve, the perpendicular distance ''v'' from the chord to the curve (usually at the chord midpoint) is called a ''versine'' measurement. For a straight line, the versine of any chord is zero, so this measurement characterizes the straightness of the curve. In the
limit Limit or Limits may refer to: Arts and media * ''Limit'' (manga), a manga by Keiko Suenobu * ''Limit'' (film), a South Korean film * Limit (music), a way to characterize harmony * "Limit" (song), a 2016 single by Luna Sea * "Limits", a 2019 ...
as the chord length ''L'' goes to zero, the ratio goes to the instantaneous
curvature In mathematics, curvature is any of several strongly related concepts in geometry. Intuitively, the curvature is the amount by which a curve deviates from being a straight line, or a surface deviates from being a plane. For curves, the canonic ...
. This usage is especially common in
rail transport Rail transport (also known as train transport) is a means of transport that transfers passengers and goods on wheeled vehicles running on rails, which are incorporated in tracks. In contrast to road transport, where the vehicles run on a p ...
, where it describes measurements of the straightness of the
rail tracks A railway track (British English and UIC terminology) or railroad track (American English), also known as permanent way or simply track, is the structure on a railway or railroad consisting of the rails, fasteners, railroad ties (sleepers, ...
and it is the basis of the
Hallade method The Hallade method, devised by Frenchman Emile Hallade, is a method used in track geometry for surveying, designing and setting out curves in railway track. It involves measuring the offset of a string line from the outside of a curve at the centr ...
for
rail surveying Surveying or land surveying is the technique, profession, art, and science of determining the land, terrestrial Two-dimensional space#In geometry, two-dimensional or Three-dimensional space#In Euclidean geometry, three-dimensional positions of ...
. The term ''
sagitta Sagitta is a dim but distinctive constellation in the northern sky. Its name is Latin for 'arrow', not to be confused with the significantly larger constellation Sagittarius 'the archer'. It was included among the 48 constellations listed by t ...
'' (often abbreviated ''sag'') is used similarly in
optics Optics is the branch of physics that studies the behaviour and properties of light, including its interactions with matter and the construction of instruments that use or detect it. Optics usually describes the behaviour of visible, ultraviole ...
, for describing the surfaces of
lens A lens is a transmissive optical device which focuses or disperses a light beam by means of refraction. A simple lens consists of a single piece of transparent material, while a compound lens consists of several simple lenses (''elements''), ...
es and
mirror A mirror or looking glass is an object that Reflection (physics), reflects an image. Light that bounces off a mirror will show an image of whatever is in front of it, when focused through the lens of the eye or a camera. Mirrors reverse the ...
s.


See also

*
Trigonometric identities 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 ...
*
Exsecant and excosecant The exsecant (exsec, exs) and excosecant (excosec, excsc, exc) are trigonometric functions defined in terms of the secant (trigonometry), secant and cosecant functions. They used to be important in fields such as surveying, railway engineering, ci ...
*
Versiera In mathematics, the witch of Agnesi () is a cubic plane curve defined from two diametrically opposite points of a circle. It gets its name from Italian mathematician Maria Gaetana Agnesi, and from a mistranslation of an Italian word for a sail ...
(
Witch of Agnesi In mathematics, the witch of Agnesi () is a cubic plane curve defined from two diametrically opposite points of a circle. It gets its name from Italian mathematician Maria Gaetana Agnesi, and from a mistranslation of an Italian word for a sail ...
) *
Exponential minus 1 The exponential function is a mathematical function denoted by f(x)=\exp(x) or e^x (where the argument is written as an exponent). Unless otherwise specified, the term generally refers to the positive-valued function of a real variable, a ...
*
Natural logarithm plus 1 The natural logarithm of a number is its logarithm to the base of the mathematical constant , which is an irrational and transcendental number approximately equal to . The natural logarithm of is generally written as , , or sometimes, if ...


Notes


References


Further reading

*


External links

*
Trigonometric Functions
at GeoGebra.org {{Trigonometric and hyperbolic functions Trigonometric functions