The versine or versed sine is a
trigonometric function found in some of the earliest (
Sanskrit ''Aryabhatia'',
[The Ä€ryabhaá¹Ä«ya by Ä€ryabhaá¹a]
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
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 sin
2 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 sin
2 = 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 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 RÃos
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 sin
2(''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.
:
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:
:
:
:
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
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
This formula was known to the Chinese mathematician
Shen Kuo
Shen Kuo (; 1031–1095) or Shen Gua, courtesy name Cunzhong (å˜ä¸) 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
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 Functionsat GeoGebra.org
{{Trigonometric and hyperbolic functions
Trigonometric functions