Exsecant

The exsecant (exsec, exs) and excosecant (excosec, excsc, exc) are trigonometric functions defined in terms of the secant and cosecant functions. They were important in fields such as surveying, railway engineering, civil engineering, astronomy, and spherical trigonometry and could help improve accuracy, but are rarely used today except to simplify some calculations.

A unit circle with trigonometric functions.

Exsecant

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.

The exsecant,[1][2][3][4][5][6][7] also known as exterior, external,[8][9][10] outward or outer secant and abbreviated as exsec[1][5][6][7][8][9][11][12][13] or exs,[14] is a trigonometric function defined in terms of the secant function sec(θ):[5][13][15]

\operatorname{exsec}(\theta) = \sec(\theta) - 1 = \frac{1}{\cos(\theta)} - 1.[5][6][7][8][9][11][12][13][15]

The name exsecant can be understood from a graphical construction of the various trigonometric functions from a unit circle, such as was used historically. sec(θ) is the secant line OE, and the exsecant is the portion DE of this secant that lies exterior to the circle (ex is Latin for out of).

Excosecant

exsecant (blue) and excosecant (green)

A related function is the excosecant[16] or coexsecant,[17][18] also known as exterior, external,[10] outward or outer cosecant and abbreviated as excosec, coexsec,[18] excsc[16] or exc,[14] the exsecant of the complementary angle:

\operatorname{excsc}(\theta) = \operatorname{exsec}\left(\frac{\pi}{2} - \theta\right) = \csc(\theta) - 1 = \frac{1}{\sin(\theta)} - 1.[16]

Usage

Once important in fields such as surveying,[6] railway engineering (for example to lay out railroad curves and superelevation), civil engineering, astronomy, and spherical trigonometry, the exsecant function is now little-used.[6][15] Mainly, this is because the availability of calculators and computers has removed the need for trigonometric tables of specialized functions such as this one.[6]

The reason to define a special function for the exsecant is similar to the rationale for the versine: for small angles θ, the sec(θ) function approaches one, and so using the above formula for the exsecant will involve the subtraction of two nearly equal quantities and exacerbate round-off 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:

\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)[2][3][10]

or

\operatorname{excsc}(\theta) = \frac{1-\sin(\theta)}{\sin(\theta)} = \frac{\operatorname{coversin}(\theta)}{\sin(\theta)} = \operatorname{coversin}(\theta) \csc(\theta) = 2 \left(\cos\left(\frac{\theta}{2}\right)\right)^2 \csc(\theta).\ [10]

Prior to the availability of computers, this would require time-consuming multiplications.

Mathematical identities

Derivatives

\frac{\mathrm{d}}{\mathrm{d}\theta}\operatorname{exsec}(\theta) = \text{tg}\theta\sec(\theta) = \frac{\sin(\theta)}{(\cos (\theta))^2}[13]
\frac{\mathrm{d}}{\mathrm{d}\theta}\operatorname{excsc}(\theta) = -\text{ctg}\theta\csc(\theta) = \frac{-\cos(\theta)}{(\sin(\theta))^2}

Integrals

\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[13]

Inverse functions

The inverse functions arcexsecant[18] (arcexsec,[18] aexsec,[19][20] aexs, exsec−1) and arcexcosecant (arcexcosec, arcexcsc, aexcsc, aexc, excsc−1) exist as well:

\operatorname{arcexsec}(y) = \arcsec(y+1) = \arccos\left(\frac{1}{y+1}\right) = \arctan(\sqrt{y^2+2y})[18][19][20] (for y ≤ −2 or y ≥ 0)[18]
\operatorname{arcexcsc}(y) = \arccsc(y+1) = \arcsin\left(\frac{1}{y+1}\right)\,

Other properties

The exsecant and excosecant functions can be extended into the complex plane.[13]

\operatorname{exsec}(\theta) = \tan(\theta) \tan\left(\frac{\theta}{2}\right)[15]

See also

References

  1. 1 2 Allen, Calvin Frank (1894) [1889]. Railroad Curves and Earthwork. New York, USA; London, UK: Spon & Chamberlain; E. & F. Spon, Ltd. Retrieved 2015-11-16.
  2. 1 2 Nagle, James C. (1897). "IV.138.-165.: Transition Curves; Table XIII.: Natural Versines and Exsecants". Field Manual for Railroad Engineers (1 ed.). New York, USA: John Wiley and Sons, Chapman and Hall, Limited. pp. 110–142;332–354. Retrieved 2015-11-16.
  3. 1 2 "Field Manual for Railroad Engineers" (PDF). The Engineer (Review): 540. 1897-12-03. Archived (PDF) from the original on 2015-11-17. Retrieved 2015-11-17.
  4. ↑ Boyer, Carl Benjamin (1969) [1959]. "5: Commentary on the Paper of E. J. Dijksterhuis (The Origins of Classical Mechanics from Aristotle to Newton)". In Clagett, Marshall. Critical Problems in the History of Science (3 ed.). Madison, Milwaukee, and London: University of Wisconsin Press, Ltd. pp. 185–190. ISBN 0-299-01874-1. LCCN 59-5304. 9780299018740. Retrieved 2015-11-16.
  5. 1 2 3 4 Zucker, Ruth (1983) [June 1964]. "Chapter 4.3.147: Elementary Transcendental Functions - Circular functions". In Abramowitz, Milton; Stegun, Irene Ann. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Applied Mathematics Series 55 (Ninth reprint with additional corrections of tenth original printing with corrections (December 1972); first ed.). Washington D.C., USA; New York, USA: United States Department of Commerce, National Bureau of Standards; Dover Publications. p. 78. ISBN 0-486-61272-4. LCCN 64-60036. MR 0167642. ISBN 978-0-486-61272-0. LCCN 65-12253.
  6. 1 2 3 4 5 6 Calvert, James B. (2007-09-14) [2004]. "Trigonometry". Archived from the original on 2007-10-02. Retrieved 2015-11-08.
  7. 1 2 3 Tapson, Frank (2004). "Background Notes on Measures: Angles". 1.4. Cleave Books. Archived from the original on 2007-02-09. Retrieved 2015-11-12.
  8. 1 2 3 Cajori, Florian (1952) [1929]. A History of Mathematical Notations 2 (2 (3rd corrected printing of 1929 issue) ed.). Chicago, USA: Open court publishing company. p. 173. ISBN 978-1-60206-714-1. 1602067147. Retrieved 2015-11-11. (NB. ISBN and link for reprint of 2nd edition by Cosimo, Inc., New York, USA, 2013.)
  9. 1 2 3 Swanson, Todd; Andersen, Janet; Keeley, Robert (1999). "5 (Trigonometric Functions)". Precalculus: A Study of Functions and Their Applications (PDF). Harcourt Brace & Company. p. 344. Archived (PDF) from the original on 2003-06-17. Retrieved 2015-11-12.
  10. 1 2 3 4 Gottschalk, Walter Helbig (2002). 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). PVD RI, GG80. Archived (PDF) from the original on 2013-09-25. Retrieved 2015-11-17.
  11. 1 2 Kenyon, Alfred Monroe; Ingold, Louis (1913). Trigonometry. New York, USA: The Macmillan Company. p. 5. Retrieved 2015-12-08.
  12. 1 2 Hudson, Ralph Gorton; Lipka, Joseph (1917). A Manual of Mathematics. New York, USA: John Wiley & sons. p. 68. Retrieved 2015-12-08.
  13. 1 2 3 4 5 6 Weisstein, Eric W. (2015) [2005]. "Exsecant". MathWorld. Wolfram Research, Inc. Archived from the original on 2005-11-29. Retrieved 2015-11-05.
  14. 1 2 Shaneyfelt, Ted V. "德博士的 Notes About Circles, ज्य, & कोज्य: What in the world is a hacovercosine?". Hilo, Hawaii: University of Hawaii. Archived from the original on 2015-09-19. Retrieved 2015-11-08.
  15. 1 2 3 4 Oldham, Keith B.; Myland, Jan C.; Spanier, Jerome (2009) [1987]. "33.13. The Secant sec(x) and Cosecant csc(x) functions - Cognate functions". An Atlas of Functions: with Equator, the Atlas Function Calculator (2 ed.). Springer Science+Business Media, LLC. p. 336. doi:10.1007/978-0-387-48807-3. ISBN 978-0-387-48806-6. LCCN 2008937525.
  16. 1 2 3 Weisstein, Eric W. (2015) [2013]. "Excosecant". MathWorld. Wolfram Research, Inc. Archived from the original on 2014-03-26. Retrieved 2015-11-05.
  17. ↑ 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 (2 (corrected) ed.). New York, USA; London, UK: D. Van Nostrand Company; Constable and Company, Ltd. Retrieved 2015-11-16.
  18. 1 2 3 4 5 6 van Vlijmen, Oscar (2005-12-28) [2003]. "Goniology". Eenheden, constanten en conversies. Archived from the original on 2009-10-28. Retrieved 2015-11-28.
  19. 1 2 Simpson, David G. (2001-11-08). "AUXTRIG" (Fortran 90 source code). Greenbelt, Maryland, USA: NASA Goddard Space Flight Center. Archived from the original on 2008-06-16. Retrieved 2015-10-26.
  20. 1 2 van den Doel, Kees (2010-01-25). "jass.utils Class Fmath". JASS - Java Audio Synthesis System. 1.25. Archived from the original on 2007-09-02. Retrieved 2015-10-26.
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