Cofunction

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"):

 ${\displaystyle \sin \left({\frac {\pi }{2}}-A\right)=\cos(A)}$[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] ${\displaystyle \operatorname {cvc} \left({\frac {\pi }{2}}-A\right)=\operatorname {vcs} (A)}$ ${\displaystyle \operatorname {hav} \left({\frac {\pi }{2}}-A\right)=\operatorname {hcv} (A)}$ ${\displaystyle \operatorname {hcv} \left({\frac {\pi }{2}}-A\right)=\operatorname {hav} (A)}$ ${\displaystyle \operatorname {hvc} \left({\frac {\pi }{2}}-A\right)=\operatorname {hcc} (A)}$ ${\displaystyle \operatorname {hcc} \left({\frac {\pi }{2}}-A\right)=\operatorname {hvc} (A)}$ ${\displaystyle \operatorname {exs} \left({\frac {\pi }{2}}-A\right)=\operatorname {exc} (A)}$ ${\displaystyle \operatorname {exc} \left({\frac {\pi }{2}}-A\right)=\operatorname {exs} (A)}$