{{Short description|Trigonometric function paired with another}} {{about|trigonometric functions|the computer program components|Coroutine}} {{For|other uses of the prefix "co" in mathematics|dual (category theory)}} [[File:Sine cosine one period.svg|thumb|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 (pairs that sum to one right angle).<ref name="Hall_1909"/> This definition typically applies to trigonometric functions.<ref name="Aufmann_Nation_2014"/><ref name="Bales_2012"/> The prefix "co-" can be found already in Edmund Gunter's ''Canon triangulorum'' (1620).<ref name="Gunter_1620"/><ref name="Roegel_2010"/>
{{anchor|Identities}}For example, sine (Latin: ''sinus'') and cosine (Latin: ''cosinus'',<ref name="Gunter_1620"/><ref name="Roegel_2010"/> ''sinus complementi''<ref name="Gunter_1620"/><ref name="Roegel_2010"/>) are cofunctions of each other (hence the "co" in "cosine"):
{| class="wikitable" |- | {{nowrap|<math>\sin\left(\frac{\pi}{2} - A\right) = \cos(A)</math><ref name="Hall_1909"/><ref name="Bales_2012"/>}} | {{nowrap|<math>\cos\left(\frac{\pi}{2} - A\right) = \sin(A)</math><ref name="Hall_1909"/><ref name="Bales_2012"/>}} |- |}
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'',<ref name="Gunter_1620"/><ref name="Roegel_2010"/> ''tangens complementi''<ref name="Gunter_1620"/><ref name="Roegel_2010"/>):
{| class="wikitable" |- | {{nowrap|<math>\sec\left(\frac{\pi}{2} - A\right) = \csc(A)</math><ref name="Hall_1909"/><ref name="Bales_2012"/>}} | {{nowrap|<math>\csc\left(\frac{\pi}{2} - A\right) = \sec(A)</math><ref name="Hall_1909"/><ref name="Bales_2012"/>}} |- | {{nowrap|<math>\tan\left(\frac{\pi}{2} - A\right) = \cot(A)</math><ref name="Hall_1909"/><ref name="Bales_2012"/>}} | {{nowrap|<math>\cot\left(\frac{\pi}{2} - A\right) = \tan(A)</math><ref name="Hall_1909"/><ref name="Bales_2012"/>}} |- |}
These equations are also known as the '''cofunction identities'''.<ref name="Aufmann_Nation_2014"/><ref name="Bales_2012"/>
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):
{| class="wikitable" |- | {{nowrap|<math>\operatorname{ver}\left(\frac{\pi}{2} - A\right) = \operatorname{cvs}(A)</math><ref name="Weisstein_covers"/>}} | {{nowrap|<math>\operatorname{cvs}\left(\frac{\pi}{2} - A\right) = \operatorname{ver}(A)</math>}} |- | {{nowrap|<math>\operatorname{vcs}\left(\frac{\pi}{2} - A\right) = \operatorname{cvc}(A)</math><ref name="Weisstein_covercos"/>}} | {{nowrap|<math>\operatorname{cvc}\left(\frac{\pi}{2} - A\right) = \operatorname{vcs}(A)</math>}} |- | {{nowrap|<math>\operatorname{hav}\left(\frac{\pi}{2} - A\right) = \operatorname{hcv}(A)</math>}} | {{nowrap|<math>\operatorname{hcv}\left(\frac{\pi}{2} - A\right) = \operatorname{hav}(A)</math>}} |- | {{nowrap|<math>\operatorname{hvc}\left(\frac{\pi}{2} - A\right) = \operatorname{hcc}(A)</math>}} | {{nowrap|<math>\operatorname{hcc}\left(\frac{\pi}{2} - A\right) = \operatorname{hvc}(A)</math>}} |- | {{nowrap|<math>\operatorname{exs}\left(\frac{\pi}{2} - A\right) = \operatorname{exc}(A)</math>}} | {{nowrap|<math>\operatorname{exc}\left(\frac{\pi}{2} - A\right) = \operatorname{exs}(A)</math>}} |- |}
==See also== * Hyperbolic functions * Lemniscatic cosine * Jacobi elliptic cosine * Cologarithm * Covariance * List of trigonometric identities
==References== <references>
<ref name="Aufmann_Nation_2014">{{cite book |title=Algebra and Trigonometry |author-first1=Richard |author-last1=Aufmann |author-first2=Richard |author-last2=Nation |edition=8 |publisher=Cengage Learning |year=2014 |isbn=978-128596583-3 |page=528 |url=https://books.google.com/books?id=JEDAAgAAQBAJ&pg=PA528 |access-date=2017-07-28}}</ref> <ref name="Gunter_1620">{{cite book |author-first=Edmund |author-last=Gunter |author-link=Edmund Gunter |title=Canon triangulorum |date=1620}}</ref> <ref name="Roegel_2010">{{cite web |title=A reconstruction of Gunter's Canon triangulorum (1620) |editor-first=Denis |editor-last=Roegel |type=Research report |publisher=HAL |date=2010-12-06 |id=inria-00543938 |url=https://hal.inria.fr/inria-00543938/document |access-date=2017-07-28 |url-status=live |archive-url=https://web.archive.org/web/20170728192238/https://hal.inria.fr/inria-00543938/document |archive-date=2017-07-28}}</ref> <ref name="Bales_2012">{{cite web |title=5.1 The Elementary Identities |work=Precalculus |author-first=John W. |author-last=Bales |date=2012 |orig-year=2001 |url=http://jwbales.home.mindspring.com/precal/part5/part5.1.html |access-date=2017-07-30 |url-status=dead |archive-url=https://web.archive.org/web/20170730201433/http://jwbales.home.mindspring.com/precal/part5/part5.1.html |archive-date=2017-07-30 }}</ref> <ref name="Hall_1909">{{cite book |title=Trigonometry |volume=Part I: Plane Trigonometry |first1=Arthur Graham |last1=Hall |first2=Fred Goodrich |last2=Frink |date=January 1909 |chapter=Chapter II. The Acute Angle [10] Functions of complementary angles |publisher=Henry Holt and Company |location=New York |pages=11–12 |url=https://archive.org/stream/planetrigonometr00hallrich#page/n26/mode/1up}}</ref> <ref name="Weisstein_covers">{{cite web |author-first=Eric Wolfgang |author-last=Weisstein |author-link=Eric Wolfgang Weisstein |title=Coversine |work=MathWorld |publisher=Wolfram Research, Inc. |url=http://mathworld.wolfram.com/Coversine.html |access-date=2015-11-06 |url-status=live |archive-url=https://web.archive.org/web/20051127184403/http://mathworld.wolfram.com/Coversine.html |archive-date=2005-11-27}}</ref> <ref name="Weisstein_covercos">{{cite web |author-first=Eric Wolfgang |author-last=Weisstein |author-link=Eric Wolfgang Weisstein |title=Covercosine |work=MathWorld |publisher=Wolfram Research, Inc. |url=http://mathworld.wolfram.com/Covercosine.html |access-date=2015-11-06 |url-status=live |archive-url=https://web.archive.org/web/20140328110051/http://mathworld.wolfram.com/Covercosine.html |archive-date=2014-03-28}}</ref>
</references>
Category:Trigonometry