直到20世紀80年代，外正割函數在測量學 [ 7] 鐵路工程 [ 4] 橫向高度 土木工程 、天文學 和球面三角學 等領域都很重要，但現在很少使用。[ 7] [ 22] 计算机 的廣泛使用已經消除了對諸如此類的專門函數的三角表的需求。[ 7]
定義一個特殊的外正割函數的原因與定義正矢函數的理據相似：
對於小角度 θ，sec(θ) 函數值接近於1，因此使用上述的外正割公式涉及減法時會幾乎會出現兩個幾乎相等的數量的相減，這個過程可能導致導致嚴重的灾难性抵消 或數值誤差。
因此，正割函數表需要非常高的精確度才能用於外正割，這使得專用的外正割函數表非常有用。
即使使用计算机，如果使用基於餘弦的定義，對於小角度的外正割來說，浮點誤差可能會成為問題。
在這種極限情況下，更準確的公式應該使用這個恒等式：[ 2] [ 3] [ 21]
exsec
(
θ
)
=
1
−
cos
(
θ
)
cos
(
θ
)
=
versin
(
θ
)
cos
(
θ
)
=
versin
(
θ
)
sec
(
θ
)
=
2
(
sin
(
θ
2
)
)
2
sec
(
θ
)
{\displaystyle \operatorname {exsec} (\theta )={\frac {1-\cos(\theta )}{\cos(\theta )}}={\frac {\operatorname {versin} (\theta )}{\cos(\theta )}}=\operatorname {versin} (\theta )\sec(\theta )=2\left(\sin \left({\frac {\theta }{2}}\right)\right)^{2}\sec(\theta )}
在計算機出現之前，這需要相當費時的乘法運算。
伽利略·伽利莱 早在 1632 年就使用了外正割函數，但當時他將其稱為segante（意為正割 ）[ 23] [ 24] [ 25] [ 26] [ 9] [ 10] [ 11] [ 12] external secant 和其縮寫縮寫ex的用法：ex.sec 函數表 [ 27] [ 28] [ 18] [ 1]
d
d
θ
exsec
(
θ
)
=
tan
(
θ
)
sec
(
θ
)
=
sin
(
θ
)
cos
2
(
θ
)
{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} \theta }}\operatorname {exsec} (\theta )=\tan(\theta )\sec(\theta )={\frac {\sin(\theta )}{\cos ^{2}(\theta )}}}
[ 16]
∫
exsec
(
θ
)
d
θ
=
ln
[
cos
(
θ
2
)
+
sin
(
θ
2
)
]
−
ln
[
cos
(
θ
2
)
−
sin
(
θ
2
)
]
−
θ
+
C
{\displaystyle \int \operatorname {exsec} (\theta )\,\mathrm {d} \theta =\ln \left[\cos \left({\frac {\theta }{2}}\right)+\sin \left({\frac {\theta }{2}}\right)\right]-\ln \left[\cos \left({\frac {\theta }{2}}\right)-\sin \left({\frac {\theta }{2}}\right)\right]-\theta +C}
[ 16]
單位圓上的外正割等價於正割 減去餘弦 再減去正矢 外正割可由單位圓推導：
exsec
(
θ
)
=
sec
(
θ
)
−
cos
(
θ
)
−
versin
(
θ
)
.
{\displaystyle \operatorname {exsec} (\theta )=\sec(\theta )-\cos(\theta )-\operatorname {versin} (\theta ).}
外正割函数可由正切函数表示为：[ 22]
exsec
(
θ
)
=
tan
(
θ
)
tan
(
θ
2
)
.
{\displaystyle \operatorname {exsec} (\theta )=\tan(\theta )\tan \left({\frac {\theta }{2}}\right).}
外正割函數與正弦函數的關係為：
exsec
(
θ
)
=
1
1
−
(
sin
(
θ
)
)
2
−
1.
{\displaystyle \operatorname {exsec} (\theta )={\frac {1}{\sqrt {1-(\sin(\theta ))^{2}}}}-1.}
外正割函數可以擴展到複平面。[ 16]
外正割的反函數 反外正割（arcexsecant[ 29] 反函數 ，符号通常表示为arcexsec [ 4] [ 29] aexsec [ 30] [ 31] aexs 或exsec−1 。
反外正割可以定義為：
arcexsec
(
y
)
=
arcsec
(
y
+
1
)
{\displaystyle \operatorname {arcexsec} (y)=\operatorname {arcsec}(y+1)}
=
arccos
(
1
y
+
1
)
=
arctan
(
y
2
+
2
y
)
{\displaystyle \ =\arccos \left({\frac {1}{y+1}}\right)=\arctan({\sqrt {y^{2}+2y}})}
[ 29] [ 30] [ 31] y ≤ −2或y ≥ 0）[ 29]
Hall, Arthur Graham; Frink, Fred Goodrich. Review Exercises [100] Secondary Trigonometric Functions. 写于Ann Arbor, Michigan, USA. Trigonometry . Part I: Plane Trigonometry. New York, USA: Henry Holt and Company [2017-08-12 ] . Calvert, James B. Trigonometry . 2007-09-14 [2004-01-10] [2015-11-08 ] . （原始内容 存档于2007-10-02）. Lemonnier, Petro (Pierre). Genneau, Ludovicum (Ludovico); Rollin, Jacobum (Jacques) , 编. Physica generalis . Cursus Philosophicus Ad Scholarum Usum Accomodatus 3 (Collegio Harcuriano [2017-08-06 ] （拉丁语） . Thysbaert, Jan-Frans. Articulus II: De situ lineæ rectæ ad Circularem; & de mensura angulorum, quorum vertex non est in circuli centro. §1. De situ lineæ rectæ ad Circularem. Definitio II: [102]. Geometria elementaria et practica . Lovanii, e typographia academica. 1774: 30, foldout [2017-08-06 ] （拉丁语） . Searles, William Henry. Field Engineering - A Hand-book of the Theory and Practice of Railway Surveying, Location, and Construction, designed for the Class-room, Field and Office, and containing a large number of useful tables, original and selected (PDF) . New York, USA: John Wiley & Sons . 1880-03-01 [2017-08-13 ] . （原始内容存档 (PDF) 于2017-08-13）. 8th revised edition, 1887 16th edition, 1910 Cajori, Florian. A History of Mathematical Notations 2 2 (3rd corrected printing of 1929 issue). Chicago, USA: Open court publishing company [2015-11-11 ] . ISBN 978-1-60206-714-1 (NB. ISBN and link for reprint of 2nd edition by Cosimo, Inc., New York, USA, 2013.)Swanson, Todd; Andersen, Janet; Keeley, Robert. 5 (Trigonometric Functions) (PDF) . Precalculus: A Study of Functions and Their Applications. Harcourt Brace & Company [2015-11-12 ] . （原始内容 (PDF) 存档于2003-06-17）. Gottschalk, Walter Helbig. Some Quaint & Curious & Almost Forgotten Trig Functions (PDF) . Gottschalk's Gestalts - A Series Illustrating Innovative Forms of the Organization & Exposition of Mathematics 80 (Providence, Rhode Island, USA: Infinite Vistas Press). 2002 [2015-11-17 ] . PVD RI, GG80. （原始内容存档 (PDF) 于2013-09-25）. Haslett, Charles. Hackley, Charles W. , 编. The Mechanic's, Machinist's, Engineer's Practical Book of Reference: Containing tables and formulæ for use in superficial and solid mensuration; strength and weight of materials; mechanics; machinery; hydraulics, hydrodynamics; marine engines, chemistry; and miscellaneous recipes. Adapted to and for the use of all classes of practical mechanics. Together with the Engineer's Field Book: Containing formulæ for the various of running and changing lines, locating side tracks and switches, &c., &c. Tables of radii and their logarithms, natural and logarithmic versed sines and external secants, natural sines and tangents to every degree and minute of the quadrant, and logarithms from the natural numbers from 1 to 10,000 . New York, USA: James G. Gregory, successor of W. A. Townsend & Co. (Stringer & Townsend). September 1855 [2017-08-13 ] . […] Still there would be much labor of computation which may be saved by the use of tables of external secant s and versed sine s, which have been employed with great success recently by the Engineers on the Ohio and Mississippi Railroad 1856 edition Poor, Henry Varnum (编). PRACTICAL BOOK OF REFERENCE, and Engineer's Field Book. By Charles Haslett, C.E. Edited by Professor Charles W. Hackley, 1 vol. 12mo. Pp. 617. Prico $2.50. Columbia College, N. Y. Stringer & Townsend (PDF) . American Railroad Journal - Steam Navigation, Commerce, Mining, Manufacturers (Review). Second Quarto Series (J. H. Schultz & Co.). 1856-03-22, XII (12): 184 [2017-08-14 ] . Whole No. 1040, Vol. XX. （原始内容存档 于2023-10-28）. van Vlijmen, Oscar. Goniology . Eenheden, constanten en conversies. 2005-12-28 [2003] [2015-11-28 ] . （原始内容存档 于2009-10-28）. van den Doel, Kees. jass.utils Class Fmath . JASS - Java Audio Synthesis System. 1.25. 2010-01-25 [2015-10-26 ] . （原始内容存档 于2007-09-02）. 
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