Editing Exsecant (section)

{{short description|Trigonometric function defined as secant minus one}}
[[File:Exsecant and versine.png|thumb|upright=1.25|The exsecant and versine functions substitute for the expressions {{math|1=exsec&thinsp;''x'' = sec&thinsp;''x''&thinsp;&minus;&thinsp;1}} and {{math|1=vers&thinsp;''x'' = 1&thinsp;&minus;&thinsp;sec&thinsp;''x''}} which appear frequently in certain applications.{{r|cajori}}]]
[[File:Versine, chord, and exsecant as line segments.png|thumb|upright=1.25|The names exsecant, versine, chord, etc. can also be applied to line segments related to a circular arc.{{r|segments}} 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 (trigonometry)|secant]] function:

<math display=block>\operatorname{exsec} \theta = \sec\theta - 1 = \frac{1}{\cos\theta} - 1.</math>

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

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.