The exsecant (exsec, exs) and excosecant (excosec, excsc, exc) are
trigonometric functions
In mathematics, the trigonometric functions (also called circular functions, angle functions or goniometric 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 ...
defined in terms of the
secant and
cosecant
In mathematics, the trigonometric functions (also called circular functions, angle functions or goniometric 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 ...
functions. They used to be important in fields such as
surveying
Surveying or land surveying is the technique, profession, art, and science of determining the terrestrial two-dimensional or three-dimensional positions of points and the distances and angles between them. A land surveying professional is ca ...
,
railway engineering,
civil engineering
Civil engineering is a professional engineering discipline that deals with the design, construction, and maintenance of the physical and naturally built environment, including public works such as roads, bridges, canals, dams, airports, sewage ...
,
astronomy
Astronomy () is a natural science that studies astronomical object, celestial objects and phenomena. It uses mathematics, physics, and chemistry in order to explain their origin and chronology of the Universe, evolution. Objects of interest ...
, and
spherical trigonometry and could help improve accuracy, but are rarely used today except to simplify some calculations.
Exsecant
The exsecant,
(Latin: ''secans exterior''
) also known as exterior, external,
outward or outer secant and abbreviated as exsec
or exs,
is a
trigonometric function
In mathematics, the trigonometric functions (also called circular functions, angle functions or goniometric 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 ...
defined in terms of the secant function sec(''θ''):
The name ''exsecant'' can be understood from a graphical construction of the various trigonometric functions from a
unit circle
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 (0, 0) in the Cartesian coordinate system in the Eucl ...
, such as was used historically. sec(''θ'') is the
secant line , and the exsecant is the portion of this secant that lies ''exterior'' to the circle (''ex'' is
Latin
Latin (, or , ) is a classical language belonging to the Italic branch of the Indo-European languages. Latin was originally a dialect spoken in the lower Tiber area (then known as Latium) around present-day Rome, but through the power of the ...
for ''out of'').
Excosecant
A related function is the excosecant
or coexsecant,
also known as exterior, external,
outward or outer cosecant and abbreviated as excosec, coexsec,
excsc
or exc,
the exsecant of the complementary angle:
Usage
Important in fields such as
surveying
Surveying or land surveying is the technique, profession, art, and science of determining the terrestrial two-dimensional or three-dimensional positions of points and the distances and angles between them. A land surveying professional is ca ...
,
railway engineering (for example to lay out
railroad curve
Track geometry is concerned with the properties and relations of points, lines, curves, and surfaces in the three-dimensional positioning of railroad track. The term is also applied to measurements used in design, construction and maintenance of ...
s and
superelevation),
civil engineering
Civil engineering is a professional engineering discipline that deals with the design, construction, and maintenance of the physical and naturally built environment, including public works such as roads, bridges, canals, dams, airports, sewage ...
,
astronomy
Astronomy () is a natural science that studies astronomical object, celestial objects and phenomena. It uses mathematics, physics, and chemistry in order to explain their origin and chronology of the Universe, evolution. Objects of interest ...
, and
spherical trigonometry up into the 1980s, the exsecant function is now little-used.
Mainly, this is because the broad availability of
calculator
An electronic calculator is typically a portable electronic device used to perform calculations, ranging from basic arithmetic to complex mathematics.
The first solid-state electronic calculator was created in the early 1960s. Pocket-sized ...
s and
computer
A computer is a machine that can be programmed to Execution (computing), carry out sequences of arithmetic or logical operations (computation) automatically. Modern digital electronic computers can perform generic sets of operations known as C ...
s has removed the need for trigonometric tables of specialized functions such as this one.
The reason to define a special function for the exsecant is similar to the rationale for the
versine: for small
angle
In Euclidean geometry, an angle is the figure formed by two Ray (geometry), rays, called the ''Side (plane geometry), sides'' of the angle, sharing a common endpoint, called the ''vertex (geometry), vertex'' of the angle.
Angles formed by two ...
s ''θ'', the sec(''θ'') function approaches
one
1 (one, unit, unity) is a number representing a single or the only entity. 1 is also a numerical digit and represents a single unit of counting or measurement. For example, a line segment of ''unit length'' is a line segment of length 1. I ...
, and so using the above formula for the exsecant will involve the
subtraction
Subtraction is an arithmetic operation that represents the operation of removing objects from a collection. Subtraction is signified by the minus sign, . For example, in the adjacent picture, there are peaches—meaning 5 peaches with 2 taken ...
of two nearly equal quantities, resulting in
catastrophic cancellation
In numerical analysis, catastrophic cancellation is the phenomenon that subtracting good approximations to two nearby numbers may yield a very bad approximation to the difference of the original numbers.
For example, if there are two studs, one L_ ...
. Thus, a table of the secant function would need a very high accuracy to be used for the exsecant, making a specialized exsecant table useful. Even with a computer,
floating point
In computing, floating-point arithmetic (FP) is arithmetic that represents real numbers approximately, using an integer with a fixed precision, called the significand, scaled by an integer exponent of a fixed base. For example, 12.345 can be ...
errors can be problematic for exsecants of small angles, if using the cosine-based definition. A more accurate formula in this limit would be to use the identity:
or
Prior to the availability of computers, this would require time-consuming multiplications.
The exsecant function was used by
Galileo Galilei
Galileo di Vincenzo Bonaiuti de' Galilei (15 February 1564 – 8 January 1642) was an Italian astronomer, physicist and engineer, sometimes described as a polymath. Commonly referred to as Galileo, his name was pronounced (, ). He was ...
in 1632 already, although he still called it ''segante'' (meaning
secant).
The Latin term ''secans exterior'' was used since at least around 1745.
The usage of the English term ''external secant'' and the abbreviation ''ex. sec.'' can be traced back to 1855 the least, when Charles Haslett published the first known
table
Table may refer to:
* Table (furniture), a piece of furniture with a flat surface and one or more legs
* Table (landform), a flat area of land
* Table (information), a data arrangement with rows and columns
* Table (database), how the table data ...
of exsecants.
Variations such as ''ex secant'' and ''exsec'' were in use in 1880,
and ''exsecant'' was used since 1894 the least.
The terms ''coexsecant''
and ''coexsec''
can be found used as early as 1880 as well
followed by ''excosecant'' since 1909.
The function was also utilized by
Albert Einstein
Albert Einstein ( ; ; 14 March 1879 – 18 April 1955) was a German-born theoretical physicist, widely acknowledged to be one of the greatest and most influential physicists of all time. Einstein is best known for developing the theory ...
to describe the
kinetic energy
In physics, the kinetic energy of an object is the energy that it possesses due to its motion.
It is defined as the work needed to accelerate a body of a given mass from rest to its stated velocity. Having gained this energy during its accele ...
of
fermion
In particle physics, a fermion is a particle that follows Fermi–Dirac statistics. Generally, it has a half-odd-integer spin: spin , spin , etc. In addition, these particles obey the Pauli exclusion principle. Fermions include all quarks an ...
s.
Mathematical identities
Derivatives
:
:
Integrals
:
:
Inverse functions
The inverse functions arcexsecant
(arcexsec,
aexsec,
aexs, exsec
−1) and arcexcosecant (arcexcosec, arcexcsc,
aexcsc, aexc, arccoexsecant, arccoexsec, excsc
−1) exist as well:
:
(for ''y'' ≤ −2 or ''y'' ≥ 0)
Other properties
Derived from the unit circle:
The exsecant function is related to the
tangent
In geometry, the tangent line (or simply tangent) to a plane curve at a given point is the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points on the curve. More ...
function by
In analogy, the excosecant function is related to the
cotangent
In mathematics, the trigonometric functions (also called circular functions, angle functions or goniometric 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 ...
function by
The exsecant function is related to the
sine
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 oppo ...
function by
In analogy, the excosecant function is related to the
cosine function by
The exsecant and excosecant functions can be extended into the
complex plane
In mathematics, the complex plane is the plane formed by the complex numbers, with a Cartesian coordinate system such that the -axis, called the real axis, is formed by the real numbers, and the -axis, called the imaginary axis, is formed by the ...
.
:
:
:
:
:
:
See also
*
*
*
*
*
*
References
{{Trigonometric and hyperbolic functions
Trigonometric functions