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Sine and cosine are each other's cofunctions.

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]

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"):

[1][3] [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]):

[1][3] [1][3]
[1][3] [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):


See also[edit]


  1. ^ a b c d e f g Hall, Arthur Graham; Frink, Fred Goodrich (January 1909). "Chapter II. The Acute Angle [10] Functions of complementary angles". Written at Ann Arbor, Michigan, USA. Trigonometry. Part I: Plane Trigonometry. New York, USA: Henry Holt and Company / Norwood Press / J. S. Cushing Co. - Berwick & Smith Co., Norwood, Massachusetts, USA. pp. 11–12. Retrieved 2017-08-12. 
  2. ^ a b Aufmann, Richard; Nation, Richard (2014). Algebra and Trigonometry (8 ed.). Cengage Learning. p. 528. ISBN 978-128596583-3. Retrieved 2017-07-28. 
  3. ^ a b c d e f g h Bales, John W. (2012) [2001]. "5.1 The Elementary Identities". Precalculus. Archived from the original on 2017-07-30. Retrieved 2017-07-30. 
  4. ^ a b c d e Gunter, Edmund (1620). Canon triangulorum. 
  5. ^ a b c d e Roegel, Denis, ed. (2010-12-06). "A reconstruction of Gunter's Canon triangulorum (1620)" (Research report). HAL. inria-00543938. Archived from the original on 2017-07-28. Retrieved 2017-07-28. 
  6. ^ Weisstein, Eric Wolfgang. "Coversine". MathWorld. Wolfram Research, Inc. Archived from the original on 2005-11-27. Retrieved 2015-11-06. 
  7. ^ Weisstein, Eric Wolfgang. "Covercosine". MathWorld. Wolfram Research, Inc. Archived from the original on 2014-03-28. Retrieved 2015-11-06.