Exsecant

The exsecant and versine functions substitute for the expressions exsec x = sec x − 1 and vers x = 1 − sec x which appear frequently in certain applications.[1]

The names exsecant, versine, chord, etc. can also be applied to line segments related to a circular arc.[2] The length of each segment is the radius times the corresponding trigonometric function of the angle.

The external secant function (abbreviated exsecant, symbolized exsec) is a trigonometric function defined in terms of the secant function:

$${\displaystyle \operatorname {exsec} \theta =\sec \theta -1={\frac {1}{\cos \theta }}-1.}$$

It was introduced in 1855 by American civil engineer Charles Haslett, who used it in conjunction with the existing versine function, ${\displaystyle \operatorname {vers} \theta =1-\cos \theta ,}$ for designing and measuring circular sections of railroad track.[3] It was adopted by surveyors and civil engineers in the United States for railroad and road design, and since the early 20th century has sometimes been briefly mentioned in American trigonometry textbooks and general-purpose engineering manuals.[4] For completeness, a few books also defined a coexsecant or excosecant function (symbolized coexsec or excsc), ${\displaystyle \operatorname {coexsec} \theta ={}}$${\displaystyle \csc \theta -1,}$ the exsecant of the complementary angle,[5][6] though it was not used in practice. While the exsecant has occasionally found other applications, today it is obscure and mainly of historical interest.[7]

As a line segment, an external secant of a circle has one endpoint on the circumference, and then extends radially outward. The length of this segment is the radius of the circle times the trigonometric exsecant of the central angle between the segment's inner endpoint and the point of tangency for a line through the outer endpoint and tangent to the circle.

Etymology

The word secant comes from Latin for "to cut", and a general secant line "cuts" a circle, intersecting it twice; this concept dates to antiquity and can be found in Book 3 of Euclid's Elements, as used e.g. in the intersecting secants theorem. 18th century sources in Latin called any non-tangential line segment external to a circle with one endpoint on the circumference a secans exterior.[8]

The trigonometric secant, named by Thomas Fincke (1583), is more specifically based on a line segment with one endpoint at the center of a circle and the other endpoint outside the circle; the circle divides this segment into a radius and an external secant. The external secant segment was used by Galileo Galilei (1632) under the name secant.[9]

History and applications

In the 19th century, most railroad tracks were constructed out of arcs of circles, called simple curves.[10] Surveyors and civil engineers working for the railroad needed to make many repetitive trigonometrical calculations to measure and plan circular sections of track. In surveying, and more generally in practical geometry, tables of both "natural" trigonometric functions and their common logarithms were used, depending on the specific calculation. Using logarithms converts expensive multiplication of multi-digit numbers to cheaper addition, and logarithmic versions of trigonometric tables further saved labor by reducing the number of necessary table lookups.[11]

The external secant or external distance of a curved track section is the shortest distance between the track and the intersection of the tangent lines from the ends of the arc, which equals the radius times the trigonometric exsecant of half the central angle subtended by the arc, ${\displaystyle R\operatorname {exsec} {\tfrac {1}{2}}\Delta .}$[12] By comparison, the versed sine of a curved track section is the furthest distance from the long chord (the line segment between endpoints) to the track[13] – cf. Sagitta – which equals the radius times the trigonometric versine of half the central angle, ${\displaystyle R\operatorname {vers} {\tfrac {1}{2}}\Delta .}$ These are both natural quantities to measure or calculate when surveying circular arcs, which must subsequently be multiplied or divided by other quantities. Charles Haslett (1855) found that directly looking up the logarithm of the exsecant and versine saved significant effort and produced more accurate results compared to calculating the same quantity from values found in previously available trigonometric tables.[3] The same idea was adopted by other authors, such as Searles (1880).[14] By 1913 Haslett's approach was so widely adopted in the American railroad industry that, in that context, "tables of external secants and versed sines [were] more common than [were] tables of secants".[15]

In the late-19th and 20th century, railroads began using arcs of an Euler spiral as a track transition curve between straight or circular sections of differing curvature. These spiral curves can be approximately calculated using exsecants and versines.[15][16]

Solving the same types of problems is required when surveying circular sections of canals[17] and roads, and the exsecant was still used in mid-20th century books about road surveying.[18]

The exsecant has sometimes been used for other applications, such as beam theory[19] and depth sounding with a wire.[20]

In recent years, the availability of calculators and computers has removed the need for trigonometric tables of specialized functions such as this one.[21] Exsecant is generally not directly built into calculators or computing environments (though it has sometimes been included in software libraries),[22] and calculations in general are much cheaper than in the past, no longer requiring tedious manual labor.

Catastrophic cancellation for small angles

Naïvely evaluating the expressions ${\displaystyle 1-\cos \theta }$ (versine) and ${\displaystyle \sec \theta -1}$ (exsecant) is problematic for small angles where ${\displaystyle \sec \theta \approx \cos \theta \approx 1.}$ Computing the difference between two approximately equal quantities results in catastrophic cancellation: because most of the digits of each quantity are the same, they cancel in the subtraction, yielding a lower-precision result.

For example, the secant of 1° is sec 1° ≈ 1.000152, with the leading several digits wasted on zeros, while the common logarithm of the exsecant of 1° is log exsec 1° ≈ −3.817220,[23] all of whose digits are meaningful. If the logarithm of exsecant is calculated by looking up the secant in a six-place trigonometric table and then subtracting 1, the difference sec 1° − 1 ≈ 0.000152 has only 3 significant digits, and after computing the logarithm only three digits are correct, log(sec 1° − 1) ≈ −3.818156.[24] For even smaller angles loss of precision is worse.

If a table or computer implementation of the exsecant function is not available, the exsecant can be accurately computed as ${\textstyle \operatorname {exsec} \theta =\tan \theta \,\tan {\tfrac {1}{2}}\theta {\vphantom {\Big |}},}$ or using versine, ${\textstyle \operatorname {exsec} \theta =\operatorname {vers} \theta \,\sec \theta ,}$ which can itself be computed as ${\textstyle \operatorname {vers} \theta =2{\bigl (}{\sin {\tfrac {1}{2}}\theta }{\bigr )}{\vphantom {)}}^{2}{\vphantom {\Big |}}={}}$${\displaystyle \sin \theta \,\tan {\tfrac {1}{2}}\theta \,{\vphantom {\Big |}}}$; Haslett used these identities to compute his 1855 exsecant and versine tables.[25][26]

For a sufficiently small angle, a circular arc is approximately shaped like a parabola, and the versine and exsecant are approximately equal to each-other and both proportional to the square of the arclength.[27]

Mathematical identities

Inverse function

The inverse of the exsecant function, which might be symbolized arcexsec,[6] is well defined if its argument ${\displaystyle y\geq 0}$ or ${\displaystyle y\leq -2}$ and can be expressed in terms of other inverse trigonometric functions (using radians for the angle):

$${\displaystyle \operatorname {arcexsec} y=\operatorname {arcsec}(y+1)={\begin{cases}{\arctan }{\bigl (}\!{\textstyle {\sqrt {y^{2}+2y}}}\,{\bigr )}&{\text{if}}\ \ y\geq 0,\\[6mu]{\text{undefined}}&{\text{if}}\ \ {-2}<y<0,\\[4mu]\pi -{\arctan }{\bigl (}\!{\textstyle {\sqrt {y^{2}+2y}}}\,{\bigr )}&{\text{if}}\ \ y\leq {-2};\\\end{cases}}_{\vphantom {.}}}$$

the arctangent expression is well behaved for small angles.[28]

Calculus

While historical uses of the exsecant did not explicitly involve calculus, its derivative and antiderivative (for x in radians) are:[29]

$${\displaystyle {\begin{aligned}{\frac {\mathrm {d} }{\mathrm {d} x}}\operatorname {exsec} x&=\tan x\,\sec x,\\[10mu]\int \operatorname {exsec} x\,\mathrm {d} x&=\ln {\bigl |}\sec x+\tan x{\bigr |}-x+C,{\vphantom {\int _{|}}}\end{aligned}}}$$

where ln is the natural logarithm. See also Integral of the secant function.

Double angle identity

The exsecant of twice an angle is:[6]

$${\displaystyle \operatorname {exsec} 2\theta ={\frac {2\sin ^{2}\theta }{1-2\sin ^{2}\theta }}.}$$

See also

• Chord (geometry) – A line segment with endpoints on the circumference of a circle, historically used trigonometrically
• Exponential minus 1 – The function ${\displaystyle x\mapsto e^{x}-1,}$ also used to improve precision for small inputs

Notes and references

1. Cajori, Florian (1929). A History of Mathematical Notations. Vol. 2. Chicago: Open Court. §527. "Less common trigonometric functions", pp. 171–172.
2. The original conception of trigonometric functions was as line segments, but this was gradually replaced during the 18th and 19th century by their conception as length ratios between sides of a right triangle or abstract functions; when the exsecant was introduced, in the mid 19th century, both concepts were still common.
   Bressoud, David (2010). "Historical Reflections on Teaching Trigonometry" (PDF). Mathematics Teacher. 104 (2): 106–112. doi:10.5951/MT.104.2.0106.

   Van Sickle, Jenna (2011). "The history of one definition: Teaching trigonometry in the US before 1900". International Journal for the History of Mathematics Education. 6 (2): 55–70.

3. Haslett, Charles (1855). "The Engineer's Field Book". In Hackley, Charles W. (ed.). The Mechanic's, Machinist's, and Engineer's Practical Book of Reference; Together with the Engineer's Field Book. New York: James G. Gregory. pp. 371–512.
   As the book's editor Charles W. Hackley explains in the preface, "The use of the more common trigonometric functions, to wit, sines, cosines, tangents, and cotangents, which ordinary tables furnish, is not well adapted to the peculiar problems which are presented in the construction of Railroad curves. [...] Still there would be much labor of computation which may be saved by the use of tables of external secants and versed sines, which have been employed with great success recently by the Engineers on the Ohio and Mississippi Railroad, and which, with the formulas and rules necessary for their application to the laying down of curves, drawn up by Mr. Haslett, one of the Engineers of that Road, are now for the first time given to the public." (pp. vi–vii)
   Charles Haslett continues in his preface to the Engineer's Field Book: "Experience has shown, that versed sines and external secants as frequently enter into calculations on curves as sines and tangents; and by their use, as illustrated in the examples given in this work, it is believed that many of the rules in general use are much simplified, and many calculations concerning curves and running lines made less intricate, and results obtained with more accuracy and far less trouble, than by any methods laid down in works of this kind. [...] In addition to the tables generally found in books of this kind, the author has prepared, with great labor, a Table of Natural and Logarithmic Versed Sines and External Secants, calculated to degrees, for every minute; also, a Table of Radii and their Logarithms, from 1° to 60°." (pp. 373–374)

   Review: Poor, Henry Varnum, ed. (1856-03-22). "Practical Book of Reference, and Engineer's Field Book. By Charles Haslett". American Railroad Journal (Review). Second Quarto Series. XII (12): 184. Whole No. 1040, Vol. XXIX.

4. Kenyon, Alfred Monroe; Ingold, Louis (1913). Trigonometry. New York: The Macmillan Company. p. 5.
   Hudson, Ralph Gorton; Lipka, Joseph (1917). A Manual of Mathematics. New York: John Wiley & Sons. p. 68.
   McNeese, Donald C.; Hoag, Albert L. (1957). Engineering and Technical Handbook. Englewood Cliffs, NJ: Prentice-Hall. pp. 147, 315–325 (table 41). LCCN 57-6690.

   Zucker, Ruth (1964). "4.3.147: Elementary Transcendental Functions - Circular functions". In Abramowitz, Milton; Stegun, Irene A. (eds.). Handbook of Mathematical Functions. Washington, D.C.: National Bureau of Standards. p. 78. LCCN 64-60036.

5. Bohannan, Rosser Daniel (1904) [1903]. "$131. The Versed Sine, Exsecant and Coexsecant. §132. Exercises". Plane Trigonometry. Boston: Allyn and Bacon. pp. 235–236.
6. Hall, Arthur Graham; Frink, Fred Goodrich (1909). "Review Exercises". Plane Trigonometry. New York: Henry Holt and Company. § "Secondary Trigonometric Functions", pp. 125–127.
7. Oldham, Keith B.; Myland, Jan C.; Spanier, Jerome (2009) [1987]. An Atlas of Functions (2nd ed.). Springer. Ch. 33, "The Secant sec(x) and Cosecant csc(x) functions", §33.13, p. 336. doi:10.1007/978-0-387-48807-3. ISBN 978-0-387-48806-6. Not appearing elsewhere in the Atlas [...] is the archaic exsecant function [...].
8. Patu, Andræâ-Claudio (André Claude); Le Tort, Bartholomæus (1745). Rivard, Franciscus (Dominique-François) [in French] (ed.). Theses Mathematicæ De Mathesi Generatim (in Latin). Paris: Ph. N. Lottin. p. 6.
   Lemonnier, Petro (Pierre) (1750). Genneau, Ludovicum (Ludovico); Rollin, Jacobum (Jacques) (eds.). Cursus Philosophicus Ad Scholarum Usum Accomodatus (in Latin). Vol. 3. Collegio Harcuriano (Collège d'Harcourt), Paris. pp. 303–.
   Thysbaert, Jan-Frans (1774). "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 (in Latin). Lovanii, e typographia academica. p. 30, foldout.

   van Haecht, Joannes (1784). "Articulus III: De secantibus circuli: Corollarium III: [109]". Geometria elementaria et practica: quam in usum auditorum (in Latin). Lovanii, e typographia academica. p. 24, foldout.

9. Galileo used the Italian segante.
   Galilei, Galileo (1632). Dialogo di Galileo Galilei sopra i due massimi sistemi del mondo Tolemaico e Copernicano [Dialogue on the Two Chief World Systems, Ptolemaic and Copernican] (in Italian).
   Galilei, Galileo (1997) [1632]. Finocchiaro, Maurice A. (ed.). Galileo on the World Systems: A New Abridged Translation and Guide. University of California Press. pp. 184 (n130), 184 (n135), 192 (n158). ISBN 9780520918221. Galileo's word is segante (meaning secant), but he clearly intends exsecant; an exsecant is defined as the part of a secant external to the circle and thus between the circumference and the tangent.

   Finocchiaro, Maurice A. (2003). "Physical-Mathematical Reasoning: Galileo on the Extruding Power of Terrestrial Rotation". Synthese. 134 (1–2, Logic and Mathematical Reasoning): 217–244. doi:10.1023/A:1022143816001. JSTOR 20117331.

10. Allen, Calvin Frank (1894) [1889]. Railroad Curves and Earthwork. New York: Spon & Chamberlain. p. 20.
11. Van Brummelen, Glen (2021). "2. Logarithms". The Doctrine of Triangles. Princeton University Press. pp. 62–109. ISBN 9780691179414.
12. Frye, Albert I. (1918) [1913]. Civil engineer's pocket-book: a reference-book for engineers, contractors and students containing rules, data, methods, formulas and tables (2nd ed.). New York: D. Van Nostrand Company. p. 211.
13. Gillespie, William M. (1853). A Manual of the Principles and Practice of Road-Making. New York: A. S. Barnes & Co. pp. 140–141.
14. Searles, William Henry (1880). Field Engineering. A hand-book of the Theory and Practice of Railway Surveying, Location, and Construction. New York: John Wiley & Sons.

    Searles, William Henry; Ives, Howard Chapin (1915) [1880]. Field Engineering: A Handbook of the Theory and Practice of Railway Surveying, Location and Construction (17th ed.). New York: John Wiley & Sons.

15. Jordan, Leonard C. (1913). The Practical Railway Spiral. New York: D. Van Nostrand Company. p. 28.
16. Thornton-Smith, G. J. (1963). "Almost Exact Closed Expressions for Computing all the Elements of the Clothoid Transition Curve". Survey Review. 17 (127): 35–44. doi:10.1179/sre.1963.17.127.35.
17. Doolittle, H. J.; Shipman, C. E. (1911). "Economic Canal Location in Uniform Countries". Papers and Discussions. Proceedings of the American Society of Civil Engineers. 37 (8): 1161–1164.
18. For example:
    Hewes, Laurence Ilsley (1942). American Highway Practice. New York: John Wiley & Sons. p. 114.
    Ives, Howard Chapin (1966) [1929]. Highway Curves (4th ed.). New York: John Wiley & Sons. LCCN 52-9033.

    Meyer, Carl F. (1969) [1949]. Route Surveying and Design (4th ed.). Scranton, PA: International Textbook Co.

19. Wilson, T. R. C. (1929). "A Graphical Method for the Solution of Certain Types of Equations". Questions and Discussions. The American Mathematical Monthly. 36 (10): 526–528. JSTOR 2299964.
20. Johnson, Harry F. (1933). "Correction for inclination of sounding wire". The International Hydrographic Review. 10 (2): 176–179.
21. Calvert, James B. (2007) [2004]. "Trigonometry". Archived from the original on 2007-10-02. Retrieved 2015-11-08.
22. Simpson, David G. (2001-11-08). "AUXTRIG" (Fortran 90 source code). Greenbelt, MD: NASA Goddard Space Flight Center. Retrieved 2015-10-26.
    van den Doel, Kees (2010-01-25). "jass.utils Class Fmath". JASS - Java Audio Synthesis System. 1.25. Retrieved 2015-10-26.

    "MIT/GNU Scheme – Scheme Arithmetic" (MIT/GNU Scheme source code). v. 12.1. Massachusetts Institute of Technology. 2023-09-01. exsec function, arith.scm lines 61–63. Retrieved 2024-04-01.

23. In a table of logarithmic exsecants such as Haslett 1855, p. 417 or Searles & Ives 1915, II. p. 135, the number given for log exsec 1° is 6.182780, the correct value plus 10, which is added to keep the entries in the table positive.
24. The incorrect digits are highlighted in red.
25. Haslett 1855, p. 415
26. Nagle, James C. (1897). "IV. Transition Curves". Field Manual for Railroad Engineers (1st ed.). New York: John Wiley and Sons. §§ 138–165, pp. 110–142; Table XIII: Natural Versines and Exsecants, pp. 332–354.

    Review: "Field Manual for Railroad Engineers. By J. C. Nagle". The Engineer (Review). 84: 540. 1897-12-03.

27. Shunk, William Findlay (1918) [1890]. The Field Engineer: A Handy Book of Practice in the Survey, Location, and Track-Work of Railroads (21st ed.). New York: D. Van Nostrand Company. p. 36.
28. "4.5 Numerical operations". MIT/GNU Scheme Documentation. v. 12.1. Massachusetts Institute of Technology. 2023-09-01. procedure: aexsec. Retrieved 2024-04-01.

    "MIT/GNU Scheme – Scheme Arithmetic" (MIT/GNU Scheme source code). v. 12.1. Massachusetts Institute of Technology. 2023-09-01. aexsec function, arith.scm lines 65–71. Retrieved 2024-04-01.

29. Weisstein, Eric W. (2015) [2005]. "Exsecant". MathWorld. Wolfram Research, Inc. Retrieved 2015-11-05.