Cofunction

This article is about trigonometric functions. For the computer program components, see Coroutine.

For other uses of the prefix "co" in mathematics, see dual (category theory).

In mathematics, a function f is cofunction of a function g if f(A) = g(B) whenever A and B are complementary angles.^[1] This definition typically applies to trigonometric functions.^[2][3] The prefix "co-" can be found already in Edmund Gunter's Canon triangulorum (1620).^[4][5]

Sine and cosine are each other's cofunctions.

For example, sine (Latin: sinus) and cosine (Latin: cosinus,^[4][5] sinus complementi^[4][5]) are cofunctions of each other (hence the "co" in "cosine"):

${\displaystyle \sin \left({\frac {\pi }{2}}-A\right)=\cos(A)}$^[1]^[3]

${\displaystyle \cos \left({\frac {\pi }{2}}-A\right)=\sin(A)}$^[1]^[3]

The same is true of secant (Latin: secans) and cosecant (Latin: cosecans, secans complementi) as well as of tangent (Latin: tangens) and cotangent (Latin: cotangens,^[4][5] tangens complementi^[4][5]):

${\displaystyle \sec \left({\frac {\pi }{2}}-A\right)=\csc(A)}$^[1]^[3]	${\displaystyle \csc \left({\frac {\pi }{2}}-A\right)=\sec(A)}$^[1]^[3]
${\displaystyle \tan \left({\frac {\pi }{2}}-A\right)=\cot(A)}$^[1]^[3]	${\displaystyle \cot \left({\frac {\pi }{2}}-A\right)=\tan(A)}$^[1]^[3]

These equations are also known as the cofunction identities.^[2][3]

This also holds true for the versine (versed sine, ver) and coversine (coversed sine, cvs), the vercosine (versed cosine, vcs) and covercosine (coversed cosine, cvc), the haversine (half-versed sine, hav) and hacoversine (half-coversed sine, hcv), the havercosine (half-versed cosine, hvc) and hacovercosine (half-coversed cosine, hcc), as well as the exsecant (external secant, exs) and excosecant (external cosecant, exc):

${\displaystyle \operatorname {ver} \left({\frac {\pi }{2}}-A\right)=\operatorname {cvs} (A)}$^[6]	${\displaystyle \operatorname {cvs} \left({\frac {\pi }{2}}-A\right)=\operatorname {ver} (A)}$
${\displaystyle \operatorname {vcs} \left({\frac {\pi }{2}}-A\right)=\operatorname {cvc} (A)}$^[7]	${\displaystyle \operatorname {cvc} \left({\frac {\pi }{2}}-A\right)=\operatorname {vcs} (A)}$
${\displaystyle \operatorname {hav} \left({\frac {\pi }{2}}-A\right)=\operatorname {hcv} (A)}$	${\displaystyle \operatorname {hcv} \left({\frac {\pi }{2}}-A\right)=\operatorname {hav} (A)}$
${\displaystyle \operatorname {hvc} \left({\frac {\pi }{2}}-A\right)=\operatorname {hcc} (A)}$	${\displaystyle \operatorname {hcc} \left({\frac {\pi }{2}}-A\right)=\operatorname {hvc} (A)}$
${\displaystyle \operatorname {exs} \left({\frac {\pi }{2}}-A\right)=\operatorname {exc} (A)}$	${\displaystyle \operatorname {exc} \left({\frac {\pi }{2}}-A\right)=\operatorname {exs} (A)}$

See also

• Hyperbolic functions
• Lemniscatic cosine
• Jacobi elliptic cosine
• Cologarithm
• Covariance
• List of trigonometric identities

References

1. Hall, Arthur Graham; Frink, Fred Goodrich (January 1909). "Chapter II. The Acute Angle [10] Functions of complementary angles". Trigonometry. Vol. Part I: Plane Trigonometry. New York: Henry Holt and Company. pp. 11–12.
2. Aufmann, Richard; Nation, Richard (2014). Algebra and Trigonometry (8 ed.). Cengage Learning. p. 528. ISBN 978-128596583-3. Retrieved 2017-07-28.
3. Bales, John W. (2012) [2001]. . Precalculus. Archived from the original on 2017-07-30. Retrieved 2017-07-30.
4. Gunter, Edmund (1620). Canon triangulorum.
5. Roegel, Denis, ed. (2010-12-06). "A reconstruction of Gunter's Canon triangulorum (1620)" (Research report). HAL. inria-00543938. from the original on 2017-07-28. Retrieved 2017-07-28.
6. Weisstein, Eric Wolfgang. "Coversine". MathWorld. Wolfram Research, Inc. from the original on 2005-11-27. Retrieved 2015-11-06.
7. Weisstein, Eric Wolfgang. "Covercosine". MathWorld. Wolfram Research, Inc. from the original on 2014-03-28. Retrieved 2015-11-06.