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Cofunction
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<p>In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, a <a href="/facts/Function_(mathematics)/CdReEKCh">function</a> <i>f</i> is cofunction of a function <i>g</i> if <i>f</i>(<i>A</i>) = <i>g</i>(<i>B</i>) whenever <i>A</i> and <i>B</i> are <a href="/facts/Complementary_angles/gFUTEQYq">complementary angles</a> (pairs that sum to one right angle). This definition typically applies to <a href="/facts/Trigonometric_functions/czej7ur0">trigonometric functions</a>. The prefix "co-" can be found already in <a href="/facts/Edmund_Gunter/wMA91AKt">Edmund Gunter</a>'s <i>Canon triangulorum</i> (1620). </p><p>For example, <a href="/facts/Sine/gQ6LXK8z">sine</a> (Latin: <i>sinus</i>) and <a href="/facts/Cosine/czej7ur0">cosine</a> (Latin: <i>cosinus</i>, <i>sinus complementi</i>) are cofunctions of each other (hence the "co" in "cosine"): </p> <table><tbody><tr><td> sin ( π 2 − A ) = cos ( A ) {\displaystyle \sin \left({\frac {\pi }{2}}-A\right)=\cos(A)} </td><td> cos ( π 2 − A ) = sin ( A ) {\displaystyle \cos \left({\frac {\pi }{2}}-A\right)=\sin(A)} </td></tr></tbody></table> <p>The same is true of <a href="/facts/Secant_(trigonometry)/czej7ur0">secant</a> (Latin: <i>secans</i>) and <a href="/facts/Cosecant/czej7ur0">cosecant</a> (Latin: <i>cosecans</i>, <i>secans complementi</i>) as well as of <a href="/facts/Tangent_(trigonometry)/czej7ur0">tangent</a> (Latin: <i>tangens</i>) and <a href="/facts/Cotangent/czej7ur0">cotangent</a> (Latin: <i>cotangens</i>, <i>tangens complementi</i>): </p> <table><tbody><tr><td> sec ( π 2 − A ) = csc ( A ) {\displaystyle \sec \left({\frac {\pi }{2}}-A\right)=\csc(A)} </td><td> csc ( π 2 − A ) = sec ( A ) {\displaystyle \csc \left({\frac {\pi }{2}}-A\right)=\sec(A)} </td></tr><tr><td> tan ( π 2 − A ) = cot ( A ) {\displaystyle \tan \left({\frac {\pi }{2}}-A\right)=\cot(A)} </td><td> cot ( π 2 − A ) = tan ( A ) {\displaystyle \cot \left({\frac {\pi }{2}}-A\right)=\tan(A)} </td></tr></tbody></table> <p>These equations are also known as the cofunction identities. </p><p>This also holds true for the <a href="/facts/Versine/MllE4ISd">versine</a> (versed sine, ver) and <a href="/facts/Coversine/MllE4ISd">coversine</a> (coversed sine, cvs), the <a href="/facts/Vercosine/MllE4ISd">vercosine</a> (versed cosine, vcs) and <a href="/facts/Covercosine/MllE4ISd">covercosine</a> (coversed cosine, cvc), the <a href="/facts/Haversine/MllE4ISd">haversine</a> (half-versed sine, hav) and <a href="/facts/Hacoversine/MllE4ISd">hacoversine</a> (half-coversed sine, hcv), the <a href="/facts/Havercosine/MllE4ISd">havercosine</a> (half-versed cosine, hvc) and <a href="/facts/Hacovercosine/MllE4ISd">hacovercosine</a> (half-coversed cosine, hcc), as well as the <a href="/facts/Exsecant/CNKrYMjC">exsecant</a> (external secant, exs) and <a href="/facts/Excosecant/CNKrYMjC">excosecant</a> (external cosecant, exc): </p> <table><tbody><tr><td> ver ( π 2 − A ) = cvs ( A ) {\displaystyle \operatorname {ver} \left({\frac {\pi }{2}}-A\right)=\operatorname {cvs} (A)} </td><td> cvs ( π 2 − A ) = ver ( A ) {\displaystyle \operatorname {cvs} \left({\frac {\pi }{2}}-A\right)=\operatorname {ver} (A)} </td></tr><tr><td> vcs ( π 2 − A ) = cvc ( A ) {\displaystyle \operatorname {vcs} \left({\frac {\pi }{2}}-A\right)=\operatorname {cvc} (A)} </td><td> cvc ( π 2 − A ) = vcs ( A ) {\displaystyle \operatorname {cvc} \left({\frac {\pi }{2}}-A\right)=\operatorname {vcs} (A)} </td></tr><tr><td> hav ( π 2 − A ) = hcv ( A ) {\displaystyle \operatorname {hav} \left({\frac {\pi }{2}}-A\right)=\operatorname {hcv} (A)} </td><td> hcv ( π 2 − A ) = hav ( A ) {\displaystyle \operatorname {hcv} \left({\frac {\pi }{2}}-A\right)=\operatorname {hav} (A)} </td></tr><tr><td> hvc ( π 2 − A ) = hcc ( A ) {\displaystyle \operatorname {hvc} \left({\frac {\pi }{2}}-A\right)=\operatorname {hcc} (A)} </td><td> hcc ( π 2 − A ) = hvc ( A ) {\displaystyle \operatorname {hcc} \left({\frac {\pi }{2}}-A\right)=\operatorname {hvc} (A)} </td></tr><tr><td> exs ( π 2 − A ) = exc ( A ) {\displaystyle \operatorname {exs} \left({\frac {\pi }{2}}-A\right)=\operatorname {exc} (A)} </td><td> exc ( π 2 − A ) = exs ( A ) {\displaystyle \operatorname {exc} \left({\frac {\pi }{2}}-A\right)=\operatorname {exs} (A)} </td></tr></tbody></table>