Exsecant

The external secant function (abbreviated exsecant, symbolized exsec) is a trigonometric function defined in terms of the secant function:

exsec

⁡

θ

=

sec

⁡

θ

−

1

=

1

cos

⁡

θ

−

1.

{\displaystyle \operatorname {exsec} \theta =\sec \theta -1={\frac {1}{\cos \theta }}-1.}

It was introduced in 1855 by American civil engineer Charles Haslett, who used it in conjunction with the existing versine function,

vers

⁡

θ

=

1

−

cos

⁡

θ

{\displaystyle \operatorname {vers} \theta =1-\cos \theta }

, for designing and measuring circular sections of railroad track. 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. For completeness, a few books also defined a coexsecant or excosecant function (symbolized coexsec or excsc),

coexsec

⁡

θ

=

{\displaystyle \operatorname {coexsec} \theta ={}}

csc

⁡

θ

−

1

{\displaystyle \csc \theta -1}

, the exsecant of the complementary angle, though it was not used in practice. While the exsecant has occasionally found other applications, today it is obscure and mainly of historical interest.

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.

Etymology

The word secant comes from Latin for "to cut", and a general secant line "cuts" a circle, intersecting it twice; this concept dates to antiquity and can be found in Book 3 of Euclid's Elements, as used e.g. in the intersecting secants theorem. 18th century sources in Latin called any non-tangential line segment external to a circle with one endpoint on the circumference a secans exterior.