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).[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"):
sin ( π 2 − A ) = cos ( A ) {\displaystyle \sin \left({\frac {\pi }{2}}-A\right)=\cos(A)} [1] [3] cos ( π 2 − A ) = sin ( A ) {\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] ):
sec ( π 2 − A ) = csc ( A ) {\displaystyle \sec \left({\frac {\pi }{2}}-A\right)=\csc(A)} [1] [3] csc ( π 2 − A ) = sec ( A ) {\displaystyle \csc \left({\frac {\pi }{2}}-A\right)=\sec(A)} [1] [3] tan ( π 2 − A ) = cot ( A ) {\displaystyle \tan \left({\frac {\pi }{2}}-A\right)=\cot(A)} [1] [3] cot ( π 2 − A ) = tan ( A ) {\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):
ver ( π 2 − A ) = cvs ( A ) {\displaystyle \operatorname {ver} \left({\frac {\pi }{2}}-A\right)=\operatorname {cvs} (A)} [6] cvs ( π 2 − A ) = ver ( A ) {\displaystyle \operatorname {cvs} \left({\frac {\pi }{2}}-A\right)=\operatorname {ver} (A)} vcs ( π 2 − A ) = cvc ( A ) {\displaystyle \operatorname {vcs} \left({\frac {\pi }{2}}-A\right)=\operatorname {cvc} (A)} [7] cvc ( π 2 − A ) = vcs ( A ) {\displaystyle \operatorname {cvc} \left({\frac {\pi }{2}}-A\right)=\operatorname {vcs} (A)} hav ( π 2 − A ) = hcv ( A ) {\displaystyle \operatorname {hav} \left({\frac {\pi }{2}}-A\right)=\operatorname {hcv} (A)} hcv ( π 2 − A ) = hav ( A ) {\displaystyle \operatorname {hcv} \left({\frac {\pi }{2}}-A\right)=\operatorname {hav} (A)} hvc ( π 2 − A ) = hcc ( A ) {\displaystyle \operatorname {hvc} \left({\frac {\pi }{2}}-A\right)=\operatorname {hcc} (A)} hcc ( π 2 − A ) = hvc ( A ) {\displaystyle \operatorname {hcc} \left({\frac {\pi }{2}}-A\right)=\operatorname {hvc} (A)} exs ( π 2 − A ) = exc ( A ) {\displaystyle \operatorname {exs} \left({\frac {\pi }{2}}-A\right)=\operatorname {exc} (A)} exc ( π 2 − A ) = exs ( A ) {\displaystyle \operatorname {exc} \left({\frac {\pi }{2}}-A\right)=\operatorname {exs} (A)}
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