|Laws and theorems|
The exsecant (exsec, exs) and excosecant (excosec, excsc, exc) are trigonometric functions defined in terms of the secant and cosecant functions. They used to be 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.
The exsecant,         (Latin: secans exterior    ) also known as exterior, external,     outward or outer secant and abbreviated as exsec     or exs,  is a trigonometric function defined in terms of the secant function sec(θ): 
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).
A related function is the excosecant   or coexsecant,    also known as exterior, external,  outward or outer cosecant and abbreviated as excosec, coexsec,    excsc   or exc,  the exsecant of the complementary angle:
Important in fields such as surveying,  railway engineering  (for example to lay out railroad curves and superelevation), civil engineering, astronomy, and spherical trigonometry up into the 1980s, the exsecant function is now little-used.   Mainly, this is because the broad availability of calculators and computers has removed the need for trigonometric tables of specialized functions such as this one. 
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, resulting in catastrophic cancellation. 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, if using the cosine-based definition. A more accurate formula in this limit would be to use the identity:
Prior to the availability of computers, this would require time-consuming multiplications.
The exsecant function was used by Galileo Galilei in 1632 already, although he still called it segante (meaning secant).     The Latin term secans exterior was used since at least around 1745.     The usage of the English term external secant and the abbreviation ex. sec. can be traced back to 1855 the least, when Charles Haslett published the first known table of exsecants.   Variations such as ex secant and exsec were in use in 1880,  and exsecant was used since 1894 the least. 
The terms coexsecant  and coexsec  can be found used as early as 1880 as well   followed by excosecant since 1909.  The function was also utilized by Albert Einstein to describe the kinetic energy of fermions.  
The inverse functions arcexsecant  (arcexsec,   aexsec,   aexs, exsec−1) and arcexcosecant (arcexcosec, arcexcsc,  aexcsc, aexc, arccoexsecant, arccoexsec, excsc−1) exist as well:
Derived from the unit circle:
The exsecant function is related to the tangent function by
In analogy, the excosecant function is related to the cotangent function by
The exsecant function is related to the sine function by
In analogy, the excosecant function is related to the cosine function by
- Trigonometric identities – Equalities that involve trigonometric functions
- Versine – 1 minus the cosine of an angle
- Chord – Geometric line segment whose endpoints both lie on the curve
- Incircle and excircles of a triangle – Circles tangent to all three sides of a triangle
- Exponential minus 1
- Natural logarithm plus 1
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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). Retrieved 2017-08-13.
[…] 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. […] In presenting this work to the public, the Author claims for it the adaptation of a new principle in trigonometrical analysis of the formulas generally used in field calculations. 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. The examples given have all been suggested by actual practice, and will explain themselves. […] As a book for practical use in field work, it is confidently believed that this is more direct in the application of rules and facility of calculation than any work now in use. 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°. […]1856 edition
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