Exsecant

The external secant function (abbreviated exsecant, symbolized exsec) is a trigonometric function defined in terms of the secant function: \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, 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), \operatorname{coexsec} \theta = {} 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.

The external secant function (abbreviated exsecant, symbolized exsec) is a trigonometric function defined in terms of the secant function: \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, 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), \operatorname{coexsec} \theta = {} 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. 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 \textstyle \operatorname{vers} \theta = 2 \bigl({\sin \tfrac12\theta}\bigr)\vphantom)^2 = {} Haslett used these identities to compute his 1855 exsecant and versine tables. For a sufficiently small angle, a circular arc is approximately shaped like a parabola, and the versine and exsecant are approximately equal to each-other and both proportional to the square of the arclength.

Mathematical identities

Inverse function

The inverse of the exsecant function, which might be symbolized , is well defined if its argument y \geq 0 or y \leq -2 and can be expressed in terms of other inverse trigonometric functions (using radians for the angle): \operatorname{arcexsec}y = \arcsec(y+1) = \begin{cases} {\arctan}\bigl(\!{\textstyle \sqrt{y^2+2y}}~\!\bigr) & \text{if}\ \ y \geq 0, \\[6mu] \text{undefined} & \text{if}\ \ {-2} the arctangent expression is well behaved for small angles.

Calculus

While historical uses of the exsecant did not explicitly involve calculus, its derivative and antiderivative (for in radians) are: \begin{align} \frac{\mathrm{d}}{\mathrm{d}x}\operatorname{exsec} x &= \tan x\,\sec x, \\[10mu] \int\operatorname{exsec} x\,\mathrm{d}x &= \ln\bigl|{\sec x + \tan x}\bigr| - x + C,\vphantom{\int_|} \end{align} where is the natural logarithm. See also Integral of the secant function.

Double angle identity

The exsecant of twice an angle is: \operatorname{exsec} 2\theta = \frac{2 \sin^2 \theta} {1 - 2 \sin^2 \theta}.

See also

Chord (geometry) – A line segment with endpoints on the circumference of a circle, historically used trigonometrically Exponential minus 1 – The function also used to improve precision for small inputs

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