HOME

TheInfoList



OR:

The versine or versed sine is a trigonometric function found in some of the earliest ( Sanskrit ''Aryabhatia'',The Āryabhaṭīya by Āryabhaṭa
Section I) trigonometric tables. 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 of navigation.


Overview

The versine or versed sine is a trigonometric function already appearing in some of the earliest trigonometric tables. It is symbolized in formulas using the abbreviations , , , or . In Latin, it is known as the ''sinus versus'' (flipped sine), ''versinus'', ''versus'', or ''sagitta'' (arrow). Expressed in terms of common trigonometric functions 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 used historically in 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: For a vertical
chord Chord may refer to: * Chord (music), an aggregate of musical pitches sounded simultaneously ** Guitar chord a chord played on a guitar, which has a particular tuning * Chord (geometry), a line segment joining two points on a curve * 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 (the horizontal axis of a graph). In 1821, Cauchy 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 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 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 Chord may refer to: * Chord (music), an aggregate of musical pitches sounded simultaneously ** Guitar chord a chord played on a guitar, which has a particular tuning * Chord (geometry), a line segment joining two points on a curve * 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 sin2 = versin(''θ''), derived by Ptolemy, that was used to construct such tables.


Haversine

The haversine, in particular, was important in navigation because it appears in the haversine formula, which is used to reasonably accurately compute distances on an astronomic spheroid (see issues with the earth's radius vs. sphere) given angular positions (e.g., longitude and latitude). 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'' (notated naturally as ''hav.'' or
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 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 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 logarithms since 1995 or in a more compact method for sight reduction since 2014.


Modern uses

Whilst 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'' < ) 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 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 or raised-cosine filter. Likewise, the havercosine is used in raised-cosine distributions in probability theory and
statistics Statistics (from German language, German: ''wikt:Statistik#German, Statistik'', "description of a State (polity), state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of ...
. In the form of sin2(''θ'') the haversine of the double-angle ''Δ'' describes the relation between spreads and angles in
rational trigonometry ''Divine Proportions: Rational Trigonometry to Universal Geometry'' is a 2005 book by the mathematician Norman J. Wildberger on a proposed alternative approach to Euclidean geometry and trigonometry, called rational trigonometry. The book advocat ...
, a proposed reformulation of metrical planar and 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: : \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, where it describes measurements of the straightness of the rail tracks 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. The term '' sagitta'' (often abbreviated ''sag'') is used similarly in optics, for describing the surfaces of lenses and mirrors.


See also

* Trigonometric identities * Exsecant and excosecant * Versiera ( Witch of Agnesi) * Exponential minus 1 * Natural logarithm plus 1


Notes


References


Further reading

*


External links

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