Exsecant

The exsecant, also abbreviated exsec, is a trigonometric function Trigonometric function

In mathematics [i], the trigonometric functions are function [i]s of an angle [i]; they are im ...

defined in terms of the secant function sec: . Once important in fields such as surveying Surveying

Surveying is the technique and science of accurately determining the terrestrial or three-dimensional sp...

, astronomy Astronomy

Astronomy is the science [i] of celestial objects and phenomena [i] that originate outside the Earth's atmosphere [i] ...

, and spherical trigonometry Spherical trigonometry

[i]s on the [[sphere]...

, the exsecant function is now little-used. Mainly, this is because the availability of calculator Calculator

A calculator is a device for performing calculation [i]s. ...

s and computer Computer

A computer is a machine [i] for manipulating data [i] according to a list of instructions [i] ...

s has removed the need for trigonometric tables of specialized functions such as this one. A related function is the excosecant , the exsecant of the complementary angle: The reason to define a special function for the exsecant is similar to the rationale for the versine Versine

The versed sine, also called the versine and, in Latin [i], the sinus versus or the sagitta' ...

: for small angle Angle

An angle is the figure formed by two rays [i] sharing a common endpoint [i], called the vertex [i] ...

s θ, the sec function approaches one, and so using the above formula for the exsecant will involve the subtraction Subtraction

Subtraction is one of the four basic arithmetic [i] operations; it is essentially the opposite of addition [i] ...

of two nearly equal quantities and exacerbate roundoff errors.

Discussions

Discussion Features

Ask a question about 'Exsecant'

Start a new discussion about 'Exsecant'

Answer questions about 'Exsecant'

'Exsecant' discussion forum

Encyclopedia

The exsecant, also abbreviated exsec, is a trigonometric function Trigonometric function

In mathematics [i], the trigonometric functions are function [i]s of an angle [i]; they are im ...

defined in terms of the secant function sec:

.

Once important in fields such as surveying Surveying

Surveying is the technique and science of accurately determining the terrestrial or three-dimensional sp...

, astronomy Astronomy

Astronomy is the science [i] of celestial objects and phenomena [i] that originate outside the Earth's atmosphere [i] ...

, and spherical trigonometry Spherical trigonometry

[i]s on the [[sphere]...

, the exsecant function is now little-used. Mainly, this is because the availability of calculator Calculator

A calculator is a device for performing calculation [i]s....

s and computer Computer

A computer is a machine [i] for manipulating data [i] according to a list of instructions [i] ...

s has removed the need for trigonometric tables of specialized functions such as this one.

A related function is the excosecant , the exsecant of the complementary angle:

The reason to define a special function for the exsecant is similar to the rationale for the versine Versine

The versed sine, also called the versine and, in Latin [i], the sinus versus or the sagitta' ...

: for small angle Angle

An angle is the figure formed by two rays [i] sharing a common endpoint [i], called the vertex [i]...

s θ, the sec function approaches one, and so using the above formula for the exsecant will involve the subtraction Subtraction

Subtraction is one of the four basic arithmetic [i] operations; it is essentially the opposite of addition [i] ...

of two nearly equal quantities and exacerbate roundoff errors. 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 errors can be problematic for exsecants of small angles. A more accurate formula in this limit would be to use the identity:

.

The name exsecant can be understood from a graphical construction, at right, of the various trigonometric functions from a unit circle Unit circle

In mathematics [i], a unit circle is a circle [i] with unit [i] radius [i], i.e., a circle whose radiu ...

, such as was used historically. sec is the secant Secant line

A secant line of a curve [i] is a line that intersects two or more point [i]s on the curve. ...

OE, and the exsecant is the portion DE of this secant that lies exterior to the circle .

References

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables , p. 78.
James B. Calvert, . Retrieved Dec. 25, 2004.