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.
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. [7]
The trigonometric secant, named by Thomas Fincke (1583), is more specifically based on a line segment with one endpoint at the center of a circle and the other endpoint outside the circle; the circle divides this segment into a radius and an external secant. The external secant segment was used by Galileo Galilei (1632) under the name secant. [8]
In the 19th century, most railroad tracks were constructed out of arcs of circles, called simple curves. [9] Surveyors and civil engineers working for the railroad needed to make many repetitive trigonometrical calculations to measure and plan circular sections of track. In surveying, and more generally in practical geometry, tables of both "natural" trigonometric functions and their common logarithms were used, depending on the specific calculation. Using logarithms converts expensive multiplication of multi-digit numbers to cheaper addition, and logarithmic versions of trigonometric tables further saved labor by reducing the number of necessary table lookups. [10]
The external secant or external distance of a curved track section is the shortest distance between the track and the intersection of the tangent lines from the ends of the arc, which equals the radius times the trigonometric exsecant of half the central angle subtended by the arc, [11] By comparison, the versed sine of a curved track section is the furthest distance from the long chord (the line segment between endpoints) to the track [12] – cf. Sagitta – which equals the radius times the trigonometric versine of half the central angle, These are both natural quantities to measure or calculate when surveying circular arcs, which must subsequently be multiplied or divided by other quantities. Charles Haslett (1855) found that directly looking up the logarithm of the exsecant and versine saved significant effort and produced more accurate results compared to calculating the same quantity from values found in previously available trigonometric tables. [2] The same idea was adopted by other authors, such as Searles (1880). [13] By 1913 Haslett's approach was so widely adopted in the American railroad industry that, in that context, "tables of external secants and versed sines [were] more common than [were] tables of secants". [14]
In the late-19th and 20th century, railroads began using arcs of an Euler spiral as a track transition curve between straight or circular sections of differing curvature. These spiral curves can be approximately calculated using exsecants and versines. [14] [15]
Solving the same types of problems is required when surveying circular sections of canals [16] and roads, and the exsecant was still used in mid-20th century books about road surveying. [17]
The exsecant has sometimes been used for other applications, such as beam theory [18] and depth sounding with a wire. [19]
In recent years, the availability of calculators and computers has removed the need for trigonometric tables of specialized functions such as this one. [20] Exsecant is generally not directly built into calculators or computing environments (though it has sometimes been included in software libraries), [21] and calculations in general are much cheaper than in the past, no longer requiring tedious manual labor.
Naïvely evaluating the expressions (versine) and (exsecant) is problematic for small angles where The difference of two nearby quantities results in catastrophic cancellation: because most of the digits of each quantity are the same, they cancel in the subtraction, yielding a lower-precision result.
For example, the secant of 1° is sec 1° ≈
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
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. [26]
The inverse of the exsecant function, which might be symbolized arcexsec, [5] is well defined if or and can be expressed in terms of other inverse trigonometric functions (using radians for the angle):
the arctangent expression is well behaved for small angles. [27]
While historical uses of the exsecant did not explicitly involve calculus, its derivative and antiderivative (for x in radians) are: [28]
The exsecant of twice an angle is: [5]
Euler's formula, named after Leonhard Euler, is a mathematical formula in complex analysis that establishes the fundamental relationship between the trigonometric functions and the complex exponential function. Euler's formula states that, for any real number x, one has
In mathematics, the slope or gradient of a line is a number that describes both the direction and the steepness of the line. Slope is often denoted by the letter m; there is no clear answer to the question why the letter m is used for slope, but its earliest use in English appears in O'Brien (1844) who wrote the equation of a straight line as "y = mx + b" and it can also be found in Todhunter (1888) who wrote it as "y = mx + c".
In mathematics, the trigonometric functions are real functions which relate an angle of a right-angled triangle to ratios of two side lengths. They are widely used in all sciences that are related to geometry, such as navigation, solid mechanics, celestial mechanics, geodesy, and many others. They are among the simplest periodic functions, and as such are also widely used for studying periodic phenomena through Fourier analysis.
In mathematics, hyperbolic functions are analogues of the ordinary trigonometric functions, but defined using the hyperbola rather than the circle. Just as the points (cos t, sin t) form a circle with a unit radius, the points (cosh t, sinh t) form the right half of the unit hyperbola. Also, similarly to how the derivatives of sin(t) and cos(t) are cos(t) and –sin(t) respectively, the derivatives of sinh(t) and cosh(t) are cosh(t) and +sinh(t) respectively.
In mathematics, tables of trigonometric functions are useful in a number of areas. Before the existence of pocket calculators, trigonometric tables were essential for navigation, science and engineering. The calculation of mathematical tables was an important area of study, which led to the development of the first mechanical computing devices.
A chord of a circle is a straight line segment whose endpoints both lie on a circular arc. If a chord were to be extended infinitely on both directions into a line, the object is a secant line. The perpendicular line passing through the chord's midpoint is called sagitta.
In mathematics, the Gudermannian function relates a hyperbolic angle measure to a circular angle measure called the gudermannian of and denoted . The Gudermannian function reveals a close relationship between the circular functions and hyperbolic functions. It was introduced in the 1760s by Johann Heinrich Lambert, and later named for Christoph Gudermann who also described the relationship between circular and hyperbolic functions in 1830. The gudermannian is sometimes called the hyperbolic amplitude as a limiting case of the Jacobi elliptic amplitude when parameter
In mathematics, the inverse trigonometric functions are the inverse functions of the trigonometric functions. Specifically, they are the inverses of the sine, cosine, tangent, cotangent, secant, and cosecant functions, and are used to obtain an angle from any of the angle's trigonometric ratios. Inverse trigonometric functions are widely used in engineering, navigation, physics, and geometry.
The versine or versed sine is a trigonometric function found in some of the earliest trigonometric tables. The versine of an angle is 1 minus its cosine.
In trigonometry, tangent half-angle formulas relate the tangent of half of an angle to trigonometric functions of the entire angle. The tangent of half an angle is the stereographic projection of the circle through the point at angle radians onto the line through the angles . Among these formulas are the following:
The Pythagorean trigonometric identity, also called simply the Pythagorean identity, is an identity expressing the Pythagorean theorem in terms of trigonometric functions. Along with the sum-of-angles formulae, it is one of the basic relations between the sine and cosine functions.
In geometry, hyperbolic angle is a real number determined by the area of the corresponding hyperbolic sector of xy = 1 in Quadrant I of the Cartesian plane. The hyperbolic angle parametrises the unit hyperbola, which has hyperbolic functions as coordinates. In mathematics, hyperbolic angle is an invariant measure as it is preserved under hyperbolic rotation.
Prosthaphaeresis was an algorithm used in the late 16th century and early 17th century for approximate multiplication and division using formulas from trigonometry. For the 25 years preceding the invention of the logarithm in 1614, it was the only known generally applicable way of approximating products quickly. Its name comes from the Greek prosthen (πρόσθεν) meaning before and aphaeresis (ἀφαίρεσις), meaning taking away or subtraction.
In mathematics, sine and cosine are trigonometric functions of an angle. The sine and cosine of an acute angle are defined in the context of a right triangle: for the specified angle, its sine is the ratio of the length of the side that is opposite that angle to the length of the longest side of the triangle, and the cosine is the ratio of the length of the adjacent leg to that of the hypotenuse. For an angle , the sine and cosine functions are denoted simply as and .
The integral of secant cubed is a frequent and challenging indefinite integral of elementary calculus:
The differentiation of trigonometric functions is the mathematical process of finding the derivative of a trigonometric function, or its rate of change with respect to a variable. For example, the derivative of the sine function is written sin′(a) = cos(a), meaning that the rate of change of sin(x) at a particular angle x = a is given by the cosine of that angle.
In mathematics, a unit circle is a circle of unit radius—that is, a radius of 1. Frequently, especially in trigonometry, the unit circle is the circle of radius 1 centered at the origin in the Cartesian coordinate system in the Euclidean plane. In topology, it is often denoted as S1 because it is a one-dimensional unit n-sphere.
The table of chords, created by the Greek astronomer, geometer, and geographer Ptolemy in Egypt during the 2nd century AD, is a trigonometric table in Book I, chapter 11 of Ptolemy's Almagest, a treatise on mathematical astronomy. It is essentially equivalent to a table of values of the sine function. It was the earliest trigonometric table extensive enough for many practical purposes, including those of astronomy. Since the 8th and 9th centuries, the sine and other trigonometric functions have been used in Islamic mathematics and astronomy, reforming the production of sine tables. Khwarizmi and Habash al-Hasib later produced a set of trigonometric tables.
In calculus, the integral of the secant function can be evaluated using a variety of methods and there are multiple ways of expressing the antiderivative, all of which can be shown to be equivalent via trigonometric identities,
Not appearing elsewhere in the Atlas, and not available through Equator, is the archaic exsecant function [...].
[...] Galileo's word is segante (meaning secant), but he clearly intends exsecant; an exsecant is defined as the part of a secant external to the circle [...]Finocchiaro, Maurice A. (2003). "Physical-Mathematical Reasoning: Galileo on the Extruding Power of Terrestrial Rotation". Synthese. 134 (1–2, Logic and Mathematical Reasoning): 217–244. JSTOR 20117331.
exsec
function, arith.scm
lines 61–63. Retrieved 2024-04-01.aexsec
function, arith.scm
lines 65–71. Retrieved 2024-04-01.