Trigonometric function defined as secant minus one
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. The length of each segment is the radius times the corresponding trigonometric function of the angle.
The external secant function (exsecant, symbolized exsec) is a trigonometric function defined in terms of the secant function:
It was introduced in 1855 by American civil engineer Charles Haslett, who used it in conjunction with the existing versine function, for designing and measuring circular sections of railroad track.[2] 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.[3] For completeness, a few books also defined a coexsecant or excosecant function (symbolized coexsec or excsc), the exsecant of the complementary angle,[4][5] though it was not used in practice. While the exsecant has occasionally found other applications, today it is obscure and mainly of historical interest.[6]
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.
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The external secant function (exsecant, symbolized exsec) is a trigonometric function defined in terms of the secant function: exsec θ = sec θ − 1...
Institute of Railway Signal Engineers (IRSE) Association of American Railroads Exsecant Degree of curvature List of engineering topics List of engineers Minimum...
cosine, hvc) and hacovercosine (half-coversed cosine, hcc), as well as the exsecant (external secant, exs) and excosecant (external cosecant, exc): Hyperbolic...
cotangent, secant, and cosecant, as well as archaic functions like versine and exsecant – can be defined geometrically in terms of a unit circle, as shown at right...
{\gamma }{2}}\right)\end{aligned}}} The versine, coversine, haversine, and exsecant were used in navigation. For example, the haversine formula was used to...
(which appeared in the earliest tables), the coversine, the haversine, the exsecant and the excosecant. The list of trigonometric identities shows more relations...
coversine, covercosine, haversine, havercosine, hacoversine, hacovercosine, exsecant and excosecant. Sloane, N. J. A. (ed.). "Sequence A001113 (Decimal expansion...
direction instead of magnetism Cartography – Study and practice of making maps Exsecant – Trigonometric function defined as secant minus one International Federation...
coversine, covercosine, haversine, havercosine, hacoversine, hacovercosine, exsecant and excosecant. Rudin, Walter (1987). Real and complex analysis (3rd ed...
describing the surfaces of lenses and mirrors. Trigonometric identities Exsecant and excosecant Versiera (Witch of Agnesi) Exponential minus 1 Natural logarithm...
their calculations, or use alternate methods. Arcsine distribution Inverse exsecant Inverse versine Inverse hyperbolic functions List of integrals of inverse...
versine and 1 / cos θ − 1 {\displaystyle 1/\!\cos \theta -1} is the exsecant. M = R ( 1 − cos ( 28.65 ? R ) ) {\displaystyle M=R\left(1-\cos \left({\frac...
Exs, or EXS may refer to: Exsecant Jet2.com, a British airline This disambiguation page lists articles associated with the title Exs. If an internal link...
– see Trigonometric function atan2 cis—see Euler's formula Cofunction Exsecant Gudermannian function Inverse trigonometric functions Jyā, koti-jyā and...
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