This article is about trigonometric functions. For the computer program components, see
Coroutine .
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)}