# Exsecant

ο»Ώ
Exsecant

The exsecant, also abbreviated exsec, is a trigonometric function defined in terms of the secant function sec(&theta;):

:$operatorname\left\{exsec\right\}\left( heta\right) = sec\left( heta\right) - 1 ,$.

Once important in fields such as surveying, astronomy, and spherical trigonometry, the exsecant function is now little-used. Mainly, this is because the availability of calculators and computers has removed the need for trigonometric tables of specialized functions such as this one.

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

:$operatorname\left\{excsc\right\}\left( heta\right) = operatorname\left\{exsec\right\}\left(pi/2 - heta\right) = csc\left( heta\right) - 1 !$

The reason to define a special function for the exsecant is similar to the rationale for the versine: for small angles ΞΈ, the sec(ΞΈ) function approaches one, and so using the above formula for the exsecant will involve the subtraction 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:

:$operatorname\left\{exsec\right\}\left( heta\right) = frac\left\{1-cos\left( heta\right)\right\}\left\{cos\left( heta\right)\right\} = frac\left\{operatorname\left\{versin\right\}\left( heta\right)\right\}\left\{cos\left( heta\right)\right\} = 2 sin^2\left( heta/2\right) sec\left( heta\right)$.

Prior to the availability of computers, this would require time-consuming multiplications.

The name "exsecant" can be understood from a graphical construction, at right, of the various trigonometric functions from a unit circle, such as was used historically. sec(ΞΈ) is the secant $overline\left\{OE\right\}$, and the exsecant is the portion $overline\left\{DE\right\}$ of this secant that lies "exterior" to the circle ("ex" is Latin for "out of").

ee also

* Trigonometric identities
* Versine and Haversine

References

* M. Abramowitz and I. A. Stegun, eds., "Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables" (Dover: New York, 1972), p. 78. (See Abramowitz and Stegun.)
* James B. Calvert, [http://www.du.edu/~jcalvert/math/trig.htm Trigonometry] (2004). Retrieved 25 December 2004.

Wikimedia Foundation. 2010.

### Look at other dictionaries:

• exsecant β noun The trigonometric function sec(x) minus; 1. Abbreviation: exsec See Also: excosecant, secant β¦   Wiktionary

• List of trigonometric identities β Cosines and sines around the unit circle β¦   Wikipedia

• Versine β The versine or versed sine, versin(ΞΈ), is a trigonometric function equal to 1 β cos(ΞΈ) and 2sin2(Β½ΞΈ). It appeared in some of the earliest trigonometric tables and was once widespread, but it is now little used. There are several related functions β¦   Wikipedia

• ΠΠ΅ΡΡΠΈΠ½ΡΡ β Π Π΅Π΄ΠΊΠΎ ΠΈΡΠΏΠΎΠ»ΡΠ·ΡΠ΅ΠΌΡΠ΅ ΡΡΠΈΠ³ΠΎΠ½ΠΎΠΌΠ΅ΡΡΠΈΡΠ΅ΡΠΊΠΈΠ΅ ΡΡΠ½ΠΊΡΠΈΠΈΒ  ΡΡΠ½ΠΊΡΠΈΠΈ ΡΠ³Π»Π°, ΠΊΠΎΡΠΎΡΡΠ΅ Π² Π½Π°ΡΡΠΎΡΡΠ΅Π΅ Π²ΡΠ΅ΠΌΡ ΠΈΡΠΏΠΎΠ»ΡΠ·ΡΡΡΡΡ ΡΠ΅Π΄ΠΊΠΎ ΠΏΠΎ ΡΡΠ°Π²Π½Π΅Π½ΠΈΡ Ρ ΡΠ΅ΡΡΡΡ ΠΎΡΠ½ΠΎΠ²Π½ΡΠΌΠΈ ΡΡΠΈΠ³ΠΎΠ½ΠΎΠΌΠ΅ΡΡΠΈΡΠ΅ΡΠΊΠΈΠΌΠΈ ΡΡΠ½ΠΊΡΠΈΡΠΌΠΈ (ΡΠΈΠ½ΡΡΠΎΠΌ, ΠΊΠΎΡΠΈΠ½ΡΡΠΎΠΌ, ΡΠ°Π½Π³Π΅Π½ΡΠΎΠΌ, ΠΊΠΎΡΠ°Π½Π³Π΅Π½ΡΠΎΠΌ, ΡΠ΅ΠΊΠ°Π½ΡΠΎΠΌ ΠΈ ΠΊΠΎΡΠ΅ΠΊΠ°Π½ΡΠΎΠΌ). Π Π½ΠΈΠΌ… β¦   ΠΠΈΠΊΠΈΠΏΠ΅Π΄ΠΈΡ

• ΠΠΎΠ²Π΅ΡΡΠΈΠ½ΡΡ β Π Π΅Π΄ΠΊΠΎ ΠΈΡΠΏΠΎΠ»ΡΠ·ΡΠ΅ΠΌΡΠ΅ ΡΡΠΈΠ³ΠΎΠ½ΠΎΠΌΠ΅ΡΡΠΈΡΠ΅ΡΠΊΠΈΠ΅ ΡΡΠ½ΠΊΡΠΈΠΈΒ  ΡΡΠ½ΠΊΡΠΈΠΈ ΡΠ³Π»Π°, ΠΊΠΎΡΠΎΡΡΠ΅ Π² Π½Π°ΡΡΠΎΡΡΠ΅Π΅ Π²ΡΠ΅ΠΌΡ ΠΈΡΠΏΠΎΠ»ΡΠ·ΡΡΡΡΡ ΡΠ΅Π΄ΠΊΠΎ ΠΏΠΎ ΡΡΠ°Π²Π½Π΅Π½ΠΈΡ Ρ ΡΠ΅ΡΡΡΡ ΠΎΡΠ½ΠΎΠ²Π½ΡΠΌΠΈ ΡΡΠΈΠ³ΠΎΠ½ΠΎΠΌΠ΅ΡΡΠΈΡΠ΅ΡΠΊΠΈΠΌΠΈ ΡΡΠ½ΠΊΡΠΈΡΠΌΠΈ (ΡΠΈΠ½ΡΡΠΎΠΌ, ΠΊΠΎΡΠΈΠ½ΡΡΠΎΠΌ, ΡΠ°Π½Π³Π΅Π½ΡΠΎΠΌ, ΠΊΠΎΡΠ°Π½Π³Π΅Π½ΡΠΎΠΌ, ΡΠ΅ΠΊΠ°Π½ΡΠΎΠΌ ΠΈ ΠΊΠΎΡΠ΅ΠΊΠ°Π½ΡΠΎΠΌ). Π Π½ΠΈΠΌ… β¦   ΠΠΈΠΊΠΈΠΏΠ΅Π΄ΠΈΡ

• ΠΠΎΡΠΈΠ½ΡΡ-Π²Π΅ΡΠ·ΡΡ β Π Π΅Π΄ΠΊΠΎ ΠΈΡΠΏΠΎΠ»ΡΠ·ΡΠ΅ΠΌΡΠ΅ ΡΡΠΈΠ³ΠΎΠ½ΠΎΠΌΠ΅ΡΡΠΈΡΠ΅ΡΠΊΠΈΠ΅ ΡΡΠ½ΠΊΡΠΈΠΈΒ  ΡΡΠ½ΠΊΡΠΈΠΈ ΡΠ³Π»Π°, ΠΊΠΎΡΠΎΡΡΠ΅ Π² Π½Π°ΡΡΠΎΡΡΠ΅Π΅ Π²ΡΠ΅ΠΌΡ ΠΈΡΠΏΠΎΠ»ΡΠ·ΡΡΡΡΡ ΡΠ΅Π΄ΠΊΠΎ ΠΏΠΎ ΡΡΠ°Π²Π½Π΅Π½ΠΈΡ Ρ ΡΠ΅ΡΡΡΡ ΠΎΡΠ½ΠΎΠ²Π½ΡΠΌΠΈ ΡΡΠΈΠ³ΠΎΠ½ΠΎΠΌΠ΅ΡΡΠΈΡΠ΅ΡΠΊΠΈΠΌΠΈ ΡΡΠ½ΠΊΡΠΈΡΠΌΠΈ (ΡΠΈΠ½ΡΡΠΎΠΌ, ΠΊΠΎΡΠΈΠ½ΡΡΠΎΠΌ, ΡΠ°Π½Π³Π΅Π½ΡΠΎΠΌ, ΠΊΠΎΡΠ°Π½Π³Π΅Π½ΡΠΎΠΌ, ΡΠ΅ΠΊΠ°Π½ΡΠΎΠΌ ΠΈ ΠΊΠΎΡΠ΅ΠΊΠ°Π½ΡΠΎΠΌ). Π Π½ΠΈΠΌ… β¦   ΠΠΈΠΊΠΈΠΏΠ΅Π΄ΠΈΡ

• ΠΠΎΡΠΈΠ½ΡΡ-Π²Π΅ΡΡΡΡ β Π Π΅Π΄ΠΊΠΎ ΠΈΡΠΏΠΎΠ»ΡΠ·ΡΠ΅ΠΌΡΠ΅ ΡΡΠΈΠ³ΠΎΠ½ΠΎΠΌΠ΅ΡΡΠΈΡΠ΅ΡΠΊΠΈΠ΅ ΡΡΠ½ΠΊΡΠΈΠΈΒ  ΡΡΠ½ΠΊΡΠΈΠΈ ΡΠ³Π»Π°, ΠΊΠΎΡΠΎΡΡΠ΅ Π² Π½Π°ΡΡΠΎΡΡΠ΅Π΅ Π²ΡΠ΅ΠΌΡ ΠΈΡΠΏΠΎΠ»ΡΠ·ΡΡΡΡΡ ΡΠ΅Π΄ΠΊΠΎ ΠΏΠΎ ΡΡΠ°Π²Π½Π΅Π½ΠΈΡ Ρ ΡΠ΅ΡΡΡΡ ΠΎΡΠ½ΠΎΠ²Π½ΡΠΌΠΈ ΡΡΠΈΠ³ΠΎΠ½ΠΎΠΌΠ΅ΡΡΠΈΡΠ΅ΡΠΊΠΈΠΌΠΈ ΡΡΠ½ΠΊΡΠΈΡΠΌΠΈ (ΡΠΈΠ½ΡΡΠΎΠΌ, ΠΊΠΎΡΠΈΠ½ΡΡΠΎΠΌ, ΡΠ°Π½Π³Π΅Π½ΡΠΎΠΌ, ΠΊΠΎΡΠ°Π½Π³Π΅Π½ΡΠΎΠΌ, ΡΠ΅ΠΊΠ°Π½ΡΠΎΠΌ ΠΈ ΠΊΠΎΡΠ΅ΠΊΠ°Π½ΡΠΎΠΌ). Π Π½ΠΈΠΌ… β¦   ΠΠΈΠΊΠΈΠΏΠ΅Π΄ΠΈΡ

• ΠΠΎΡΠΈΠ½ΡΡ Π²Π΅ΡΠ·ΡΡ β Π Π΅Π΄ΠΊΠΎ ΠΈΡΠΏΠΎΠ»ΡΠ·ΡΠ΅ΠΌΡΠ΅ ΡΡΠΈΠ³ΠΎΠ½ΠΎΠΌΠ΅ΡΡΠΈΡΠ΅ΡΠΊΠΈΠ΅ ΡΡΠ½ΠΊΡΠΈΠΈΒ  ΡΡΠ½ΠΊΡΠΈΠΈ ΡΠ³Π»Π°, ΠΊΠΎΡΠΎΡΡΠ΅ Π² Π½Π°ΡΡΠΎΡΡΠ΅Π΅ Π²ΡΠ΅ΠΌΡ ΠΈΡΠΏΠΎΠ»ΡΠ·ΡΡΡΡΡ ΡΠ΅Π΄ΠΊΠΎ ΠΏΠΎ ΡΡΠ°Π²Π½Π΅Π½ΠΈΡ Ρ ΡΠ΅ΡΡΡΡ ΠΎΡΠ½ΠΎΠ²Π½ΡΠΌΠΈ ΡΡΠΈΠ³ΠΎΠ½ΠΎΠΌΠ΅ΡΡΠΈΡΠ΅ΡΠΊΠΈΠΌΠΈ ΡΡΠ½ΠΊΡΠΈΡΠΌΠΈ (ΡΠΈΠ½ΡΡΠΎΠΌ, ΠΊΠΎΡΠΈΠ½ΡΡΠΎΠΌ, ΡΠ°Π½Π³Π΅Π½ΡΠΎΠΌ, ΠΊΠΎΡΠ°Π½Π³Π΅Π½ΡΠΎΠΌ, ΡΠ΅ΠΊΠ°Π½ΡΠΎΠΌ ΠΈ ΠΊΠΎΡΠ΅ΠΊΠ°Π½ΡΠΎΠΌ). Π Π½ΠΈΠΌ… β¦   ΠΠΈΠΊΠΈΠΏΠ΅Π΄ΠΈΡ

• ΠΠΎΡΠΈΠ½ΡΡ Π²Π΅ΡΡΡΡ β Π Π΅Π΄ΠΊΠΎ ΠΈΡΠΏΠΎΠ»ΡΠ·ΡΠ΅ΠΌΡΠ΅ ΡΡΠΈΠ³ΠΎΠ½ΠΎΠΌΠ΅ΡΡΠΈΡΠ΅ΡΠΊΠΈΠ΅ ΡΡΠ½ΠΊΡΠΈΠΈΒ  ΡΡΠ½ΠΊΡΠΈΠΈ ΡΠ³Π»Π°, ΠΊΠΎΡΠΎΡΡΠ΅ Π² Π½Π°ΡΡΠΎΡΡΠ΅Π΅ Π²ΡΠ΅ΠΌΡ ΠΈΡΠΏΠΎΠ»ΡΠ·ΡΡΡΡΡ ΡΠ΅Π΄ΠΊΠΎ ΠΏΠΎ ΡΡΠ°Π²Π½Π΅Π½ΠΈΡ Ρ ΡΠ΅ΡΡΡΡ ΠΎΡΠ½ΠΎΠ²Π½ΡΠΌΠΈ ΡΡΠΈΠ³ΠΎΠ½ΠΎΠΌΠ΅ΡΡΠΈΡΠ΅ΡΠΊΠΈΠΌΠΈ ΡΡΠ½ΠΊΡΠΈΡΠΌΠΈ (ΡΠΈΠ½ΡΡΠΎΠΌ, ΠΊΠΎΡΠΈΠ½ΡΡΠΎΠΌ, ΡΠ°Π½Π³Π΅Π½ΡΠΎΠΌ, ΠΊΠΎΡΠ°Π½Π³Π΅Π½ΡΠΎΠΌ, ΡΠ΅ΠΊΠ°Π½ΡΠΎΠΌ ΠΈ ΠΊΠΎΡΠ΅ΠΊΠ°Π½ΡΠΎΠΌ). Π Π½ΠΈΠΌ… β¦   ΠΠΈΠΊΠΈΠΏΠ΅Π΄ΠΈΡ

• Π₯Π°Π²Π΅ΡΡΠΈΠ½ΡΡ β Π Π΅Π΄ΠΊΠΎ ΠΈΡΠΏΠΎΠ»ΡΠ·ΡΠ΅ΠΌΡΠ΅ ΡΡΠΈΠ³ΠΎΠ½ΠΎΠΌΠ΅ΡΡΠΈΡΠ΅ΡΠΊΠΈΠ΅ ΡΡΠ½ΠΊΡΠΈΠΈΒ  ΡΡΠ½ΠΊΡΠΈΠΈ ΡΠ³Π»Π°, ΠΊΠΎΡΠΎΡΡΠ΅ Π² Π½Π°ΡΡΠΎΡΡΠ΅Π΅ Π²ΡΠ΅ΠΌΡ ΠΈΡΠΏΠΎΠ»ΡΠ·ΡΡΡΡΡ ΡΠ΅Π΄ΠΊΠΎ ΠΏΠΎ ΡΡΠ°Π²Π½Π΅Π½ΠΈΡ Ρ ΡΠ΅ΡΡΡΡ ΠΎΡΠ½ΠΎΠ²Π½ΡΠΌΠΈ ΡΡΠΈΠ³ΠΎΠ½ΠΎΠΌΠ΅ΡΡΠΈΡΠ΅ΡΠΊΠΈΠΌΠΈ ΡΡΠ½ΠΊΡΠΈΡΠΌΠΈ (ΡΠΈΠ½ΡΡΠΎΠΌ, ΠΊΠΎΡΠΈΠ½ΡΡΠΎΠΌ, ΡΠ°Π½Π³Π΅Π½ΡΠΎΠΌ, ΠΊΠΎΡΠ°Π½Π³Π΅Π½ΡΠΎΠΌ, ΡΠ΅ΠΊΠ°Π½ΡΠΎΠΌ ΠΈ ΠΊΠΎΡΠ΅ΠΊΠ°Π½ΡΠΎΠΌ). Π Π½ΠΈΠΌ… β¦   ΠΠΈΠΊΠΈΠΏΠ΅Π΄ΠΈΡ

### Share the article and excerpts

##### Direct link
β¦ Do a right-click on the link above
and select βCopy Linkβ

We are using cookies for the best presentation of our site. Continuing to use this site, you agree with this.