Trigonometric function defined as secant minus one
The exsecant and versine functions substitute for the expressionsexsec x = sec x − 1 andvers 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.
Theexternal secant function (abbreviatedexsecant, symbolizedexsec) is atrigonometric function defined in terms of thesecant function:
It was introduced in 1855 by Americancivil engineerCharles Haslett, who used it in conjunction with the existingversine function, for designing and measuringcircular sections ofrailroad track.[3] It was adopted bysurveyors and civil engineers in the United States for railroad androad 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 acoexsecant orexcosecant function (symbolizedcoexsec orexcsc), the exsecant of thecomplementary 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 aline segment, anexternal secant of acircle 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 thepoint of tangency for a line through the outer endpoint andtangent to the circle.
The wordsecant comes from Latin for "to cut", and a generalsecant line "cuts" a circle, intersecting it twice; this concept dates to antiquity and can be found in Book 3 ofEuclid'sElements, as used e.g. in theintersecting secants theorem. 18th century sources inLatin calledany non-tangential line segment external to a circle with one endpoint on the circumference asecans exterior.[8]
The trigonometricsecant, named byThomas 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 byGalileo Galilei (1632) under the namesecant.[9]
In the 19th century, mostrailroad tracks were constructed out ofarcs of circles, calledsimple curves.[10]Surveyors andcivil 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 theircommon 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]
Theexternal secant orexternal 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 thecentral angle subtended by the arc,[12] By comparison, theversed sine of a curved track section is the furthest distance from thelongchord (the line segment between endpoints) to the track[13] – cf.Sagitta – which equals the radius times the trigonometric versine of half the central angle, 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 anEuler spiral as atrack 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 ofcanals[17] and roads, and the exsecant was still used in mid-20th century books about road surveying.[18]
In recent years, the availability ofcalculators andcomputers 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 insoftware libraries),[22] and calculations in general are much cheaper than in the past, no longer requiring tedious manual labor.
Naïvely evaluating the expressions (versine) and (exsecant) is problematic for small angles where Computing the difference between two approximately equal quantities results incatastrophic 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 of1° issec 1° ≈ 1.000152, with the leading several digits wasted on zeros, while thecommon logarithm of the exsecant of1° islog 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-placetrigonometric table and then subtracting1, the differencesec 1° − 1 ≈ 0.000152 has only 3significant 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 or using versine, which can itself be computed as; 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 aparabola, and the versine and exsecant are approximately equal to each-other and both proportional to the square of the arclength.[27]
Theinverse of the exsecant function, which might be symbolizedarcexsec,[6] is well defined if its argument or and can be expressed in terms of otherinverse trigonometric functions (usingradians for the angle):
the arctangent expression is well behaved for small angles.[28]
^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.
^abHaslett, 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 ofexternal secants andversed sines, which have been employed with great success recently by the Engineers on theOhio 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 theEngineer'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)
^abcHall, Arthur Graham; Frink, Fred Goodrich (1909)."Review Exercises".Plane Trigonometry. New York:Henry Holt and Company. § "Secondary Trigonometric Functions", pp. 125–127.
^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.ISBN978-0-387-48806-6.Not appearing elsewhere in theAtlas [...] is the archaicexsecant function [...].
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.JSTOR20117331.
^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.
^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.LCCN52-9033.
Meyer, Carl F. (1969) [1949].Route Surveying and Design (4th ed.). Scranton, PA: International Textbook Co.
^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.JSTOR2299964.
^ In a table of logarithmic exsecants such asHaslett 1855,p. 417 orSearles & Ives 1915,II. p. 135, the number given forlog exsec 1° is6.182780, the correct value plus10, which is added to keep the entries in the table positive.
