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| [[Image:Exsecant and exosecant plot.png|frame|right|exsecant (blue) and excosecant (green)]]
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| [[Image:Circle-trig6.svg|right|thumb|320px|The trigonometric functions, including the exsecant, can be constructed geometrically in terms of a unit circle centered at ''O''. The exsecant is the portion ''DE'' of the secant ''exterior'' to (''ex'') the circle.]]
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| The '''exsecant''', also abbreviated '''exsec''', is a [[trigonometric function]] defined in terms of the secant function sec(θ):
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| :<math>\operatorname{exsec}(\theta) = \sec(\theta) - 1. \,</math>
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| Once important in fields such as [[surveying]], [[astronomy]], and [[spherical trigonometry]], the exsecant function is now little-used. Mainly, this is because the availability of [[calculator]]s and [[computer]]s has removed the need for trigonometric tables of specialized functions such as this one.
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| A related function is the '''excosecant''' ('''excsc'''), the exsecant of the complementary angle:
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| :<math>\operatorname{excsc}(\theta) = \operatorname{exsec}(\pi/2 - \theta) = \csc(\theta) - 1. \!</math>
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| The reason to define a special function for the exsecant is similar to the rationale for the [[versine]]: for small [[angle]]s <var>θ</var>, the sec(<var>θ</var>) function approaches [[1 (number)|one]], and so using the above formula for the exsecant will involve the [[subtraction]] of two nearly equal quantities and exacerbate roundoff errors. Thus, a table of the secant function would need a very high accuracy to be used for the exsecant, making a specialized exsecant table useful. Even with a computer, [[floating point]] errors can be problematic for exsecants of small angles. A more accurate formula in this limit would be to use the identity:
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| :<math>\operatorname{exsec}(\theta) = \frac{1-\cos(\theta)}{\cos(\theta)}
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| = \frac{\operatorname{versin}(\theta)}{\cos(\theta)}
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| = 2 \sin^2(\theta/2) \sec(\theta).\ </math>
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| Prior to the availability of computers, this would require time-consuming multiplications.
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| The name ''exsecant'' can be understood from a graphical construction, at right, of the various trigonometric functions from a [[unit circle]], such as was used historically. sec(<var>θ</var>) is the [[secant line|secant]] <math>\overline{OE}</math>, and the exsecant is the portion <math>\overline{DE}</math> of this secant that lies ''exterior'' to the circle (''ex'' is [[Latin]] for ''out of'').
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| ==See also==
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| * [[List of trigonometric identities#Historic shorthands|Trigonometric identities]]
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| * [[Versine|Versine and haversine]]
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| ==References==
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| * M. Abramowitz and I. A. Stegun, eds., ''Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables'' (Dover: New York, 1972), p. 78. (See [[Abramowitz and Stegun]].)
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| * James B. Calvert, [http://www.du.edu/~jcalvert/math/trig.htm Trigonometry] (2004). Retrieved 25 December 2004.
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| [[Category:Trigonometry]]
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| [[Category:Elementary special functions]]
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| {{geometry-stub}}
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