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| The '''versine''' or '''versed sine''', versin(''θ''), is a [[trigonometric function]] equal to {{nowrap|1 − cos(''θ'')}} and 2sin<sup>2</sup>(½''θ''). It appeared in some of the earliest trigonometric tables and was once widespread, but it is now little-used. There are several related functions, most notably the '''haversine''', half the versine, known in the [[haversine formula]] of navigation.
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| It is also written as vers(''θ'') or ver(''θ''). In [[Latin]], it is known as the ''sinus versus'' (flipped sine) or the ''sagitta'' (arrow).
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| ==Related functions==
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| There are several other related functions:
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| * The '''versed cosine''', or '''vercosine''', written <math>\operatorname{vercosin}(\theta)</math>
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| * The '''coversed sine''', or '''coversine''', written <math>\operatorname{coversin}(\theta)</math> and sometimes abbreviated to <math>\operatorname{cvs}(\theta)</math>
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| * The '''coversed cosine''', or '''covercosine''', written <math>\operatorname{covercosin}(\theta)</math>
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| * The '''haversed sine''', or '''haversine''', written <math>\operatorname{haversin}(\theta)</math>, most famous from the [[haversine formula]] used historically in [[navigation]]
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| * The '''haversed cosine''', or '''havercosine''', written <math>\operatorname{havercosin}(\theta)</math>
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| * The '''hacoversed sine''', also called '''hacoversine''' or '''cohaversine''' and written <math>\operatorname{hacoversin}(\theta)</math>
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| * The '''hacoversed cosine''', also called '''hacovercosine''' or '''cohavercosine''' and written <math>\operatorname{hacovercosin}(\theta)</math>
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| *The '''[[exsecant]]''', written <math>\operatorname{exsec}(\theta)</math>
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| *The '''excosecant''', written <math>\operatorname{excosec}(\theta)</math>
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| [[Image:Circle-trig6.svg|right|thumb|320px|The trigonometric functions can be constructed geometrically in terms of a unit circle centered at ''O''. This figure also illustrates the reason why the versine was sometimes called the ''sagitta'', Latin for [[arrow]].<ref name=OED>{{OED|sagitta}}</ref> If the arc ''ADB'' is viewed as a "[[bow (weapon)|bow]]" and the chord ''AB'' as its "string", then the versine ''CD'' is clearly the "arrow shaft".]]
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| ==Definitions==
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| {| class="wikitable"
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| |-
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| | <math>\textrm{versin} (\theta) := 2\sin^2\!\left(\frac{\theta}{2}\right) = 1 - \cos (\theta) \,</math>
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| || [[image:Versin plot.png|400px]]
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| |-
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| | <math>\textrm{vercosin} (\theta) := 2\cos^2\!\left(\frac{\theta}{2}\right) = 1 + \cos (\theta) \,</math>
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| || [[image:Vercosin plot.png|400px]]
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| |-
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| | <math>\textrm{coversin}(\theta) := \textrm{versin}\!\left(\frac{\pi}{2} - \theta\right) = 1 - \sin(\theta) \,</math>
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| || [[image:Coversin plot.png|400px]]
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| |-
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| | <math>\textrm{covercosin}(\theta) := \textrm{vercosin}\!\left(\frac{\pi}{2} - \theta\right) = 1 + \sin(\theta) \,</math>
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| || [[image:Covercosin plot.png|400px]]
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| |-
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| | <math>\textrm{haversin}(\theta) := \frac {\textrm{versin}(\theta)} {2} = \frac{1 - \cos (\theta)}{2} \,</math>
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| || [[image:Haversin plot.png|400px]]
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| |-
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| | <math>\textrm{havercosin}(\theta) := \frac {\textrm{vercosin}(\theta)} {2} = \frac{1 + \cos (\theta)}{2} \,</math>
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| || [[image:Havercosin plot.png|400px]]
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| |-
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| | <math>\textrm{hacoversin}(\theta) := \frac {\textrm{coversin}(\theta)} {2} = \frac{1 - \sin (\theta)}{2} \,</math>
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| || [[image:Hacoversin plot.png|400px]]
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| |-
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| | <math>\textrm{hacovercosin}(\theta) := \frac {\textrm{covercosin}(\theta)} {2} = \frac{1 + \sin (\theta)}{2} \,</math>
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| || [[image:Hacovercosin plot.png|400px]]
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| |}
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| ==Derivatives and Integrals==
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| {| class="wikitable"
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| |-
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| | <math>\frac{\mathrm{d}}{\mathrm{d}x}\mathrm{versin}(x) = \sin{x}</math>
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| || <math>\int\mathrm{versin}(x) \,\mathrm{d}x = x - \sin{x} + C</math>
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| |-
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| | <math>\frac{\mathrm{d}}{\mathrm{d}x}\mathrm{vercosin}(x) = -\sin{x}</math>
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| || <math>\int\mathrm{vercosin}(x) \,\mathrm{d}x = x + \sin{x} + C</math>
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| |-
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| | <math>\frac{\mathrm{d}}{\mathrm{d}x}\mathrm{coversin}(x) = -\cos{x}</math>
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| || <math>\int\mathrm{coversin}(x) \,\mathrm{d}x = x + \cos{x} + C</math>
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| |-
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| | <math>\frac{\mathrm{d}}{\mathrm{d}x}\mathrm{covercosin}(x) = \cos{x}</math>
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| || <math>\int\mathrm{covercosin}(x) \,\mathrm{d}x = x - \cos{x} + C</math>
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| |-
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| | <math>\frac{\mathrm{d}}{\mathrm{d}x}\mathrm{haversin}(x) = \frac{\sin{x}}{2}</math>
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| || <math>\int\mathrm{haversin}(x) \,\mathrm{d}x = \frac{x - \sin{x}}{2} + C</math>
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| |-
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| | <math>\frac{\mathrm{d}}{\mathrm{d}x}\mathrm{havercosin}(x) = \frac{-\sin{x}}{2}</math>
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| || <math>\int\mathrm{havercosin}(x) \,\mathrm{d}x = \frac{x + \sin{x}}{2} + C</math>
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| |-
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| | <math>\frac{\mathrm{d}}{\mathrm{d}x}\mathrm{hacoversin}(x) = \frac{-\cos{x}}{2}</math>
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| || <math>\int\mathrm{hacoversin}(x) \,\mathrm{d}x = \frac{x + \cos{x}}{2} + C</math>
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| |-
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| | <math>\frac{\mathrm{d}}{\mathrm{d}x}\mathrm{hacovercosin}(x) = \frac{\cos{x}}{2}</math>
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| || <math>\int\mathrm{hacovercosin}(x) \,\mathrm{d}x = \frac{x - \cos{x}}{2} + C</math>
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| |}
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| ==History and applications==
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| Historically, the versed sine was considered one of the most important trigonometric functions,<ref name=boyer/><ref name=miller>{{cite web |first=J. |last=Miller |url=http://jeff560.tripod.com/v.html |title=Earliest known uses of some of the words of mathematics (v)}}</ref><ref name=Calvert/> but it has fallen from popularity in modern times due to the availability of [[computer]]s and scientific [[calculator]]s. 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.<ref name=Calvert/> Even with a computer, [[round-off error]]s make it advisable to use the sin<sup>2</sup> 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''π'',...) where it is zero—thus, one could use logarithmic tables for multiplications in formulas involving versines.
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| The haversine, in particular, was important in [[navigation]] because it appears in the [[haversine formula]], which is used to reasonably accurately compute distances on a sphere (see issues with the Earth`s radius vs. sphere) given angular positions (e.g., [[longitude]] and [[latitude]]). One could also use sin<sup>2</sup>(''θ''/2) directly, but having a table of the haversine removed the need to compute squares and square roots.<ref name=Calvert>{{cite web |first=James B. |last=Calvert |url=http://www.du.edu/~jcalvert/math/trig.htm |title=Trigonometry}}</ref> The term ''haversine'' was, apparently, coined in a navigation text for just such an application.<ref>{{OED|haversine}} Cites coinage by Prof. Jas. Inman, D. D., in his ''Navigation and Nautical Astronomy'', 3rd ed. (1835).</ref>
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| In fact, the earliest surviving table of [[sine]] (half-[[Chord (geometry)|chord]]) values (as opposed to the [[Ptolemy's table of chords|chords tabulated by Ptolemy]] and other Greek authors), calculated from the [[Surya_Siddhanta|Surya Siddhantha]] of India dated back to 3rd century BC, was a table of values for the sine and versed sine (in 3.75° increments from 0 to 90°).<ref name=boyer>{{cite book |first=Carl B. |last=Boyer |title=A History of Mathematics |edition=2nd |publisher=[[John Wiley & Sons|Wiley]] |location=New York |year=1991}}</ref> The versine appears as an intermediate step in the application of the half-angle formula sin<sup>2</sup>(''θ''/2) = versin(''θ'')/2, derived by [[Ptolemy]], that was used to construct such tables.
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| [[Image:Versine.svg|right|thumb|Sine, cosine, and versine of θ in terms of a unit circle, centered at ''O'']]
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| The ordinary ''sine'' function ([[History of trigonometric functions#Etymology|see note on etymology]]) was sometimes historically called the ''sinus rectus'' ("vertical sine"), to contrast it with the versed sine (''sinus versus'').<ref name=boyer/> The meaning of these terms is apparent if one looks at the functions in the original context for their definition, a unit circle, shown at right. For a vertical chord ''AB'' of the unit circle, the sine of the angle θ (half the subtended angle) 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'') = ''OC'' and versin(θ) = ''CD'' is the radius ''OD'' = 1. Illustrated this way, the sine is vertical (''rectus'', lit. "straight") while the versine is horizontal (''versus'', lit. "turned against, out-of-place"); both are distances from ''C'' to the circle.
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| This figure also illustrates the reason why the versine was sometimes called the ''sagitta'', Latin for [[arrow]],<ref name=OED>{{OED|sagitta}}</ref> from the Arabic usage ''sahem''<ref name=miller/> of the same meaning. This itself comes from the Indian word 'sara' (arrow) that was commonly used to refere to "[[Jyā,_koti-jyā_and_utkrama-jyā|utkrama-jya]]". If the arc ''ADB'' is viewed as a "[[bow (weapon)|bow]]" and the chord ''AB'' as its "string", then the versine ''CD'' is clearly the "arrow shaft".
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| 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).<ref name=OED/>
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| One period (0 < ''θ'' < ''π''/2) of a versine or, more commonly, a haversine waveform is also commonly used in [[signal processing]] and [[control theory]] as the shape of a [[pulse (signal processing)|pulse]] or a [[window function]], because it smoothly ([[continuous function|continuous]] in value and [[slope]]) "turns on" from [[0 (number)|zero]] to [[1 (number)|one]] (for haversine) and back to zero. In these applications, it is given yet another name: [[raised-cosine filter]] or [[Hann function]].
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| =="Versines" of arbitrary curves and chords==
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| 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 (mathematics)|limit]] as the chord length ''L'' goes to zero, the ratio 8''v''/''L''<sup>2</sup> goes to the instantaneous [[curvature]]. This usage is especially common in [[rail transport]], where it describes measurements of the straightness of the [[rail tracks]]<ref>{{cite journal |first=Bhaskaran |last=Nair |title=Track measurement systems—concepts and techniques |journal=Rail International |volume=3 |issue=3 |pages=159–166 |year=1972 |issn=0020-8442}}</ref> and it is the basis of the [[Hallade method]] for rail surveying. The term '[[sagitta (geometry)|sagitta]]' (often abbreviated ''sag'') is used similarly in [[optics]], for describing the surfaces of [[lens (optics)|lenses]] and [[mirror]]s.
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| ==See also==
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| * [[List of trigonometric identities#Historic shorthands|Trigonometric identities]]
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| * [[Exsecant]]
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| * [[sagitta (geometry)|Sagitta]]
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| == References ==
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| {{reflist}}
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| == External links ==
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| * {{MathWorld | urlname=Versine | title=Versine}}
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| * {{MathWorld | urlname=Haversine | title=Haversine}}
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| * [http://demonstrations.wolfram.com/SagittaApothemAndChord/ Sagitta, Apothem, and Chord] by [[Ed Pegg, Jr.]], [[The Wolfram Demonstrations Project]].
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| [[Category:Trigonometry]]
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| [[Category:Elementary special functions]]
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