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This article is about trigonometric functions. For the computer program components, see
Coroutine .
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).
[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"):
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)}
See also
References
^
a
b
c
d
e
f
g Hall, Arthur Graham; Frink, Fred Goodrich (January 1909). "Chapter II. The Acute Angle [10] Functions of complementary angles".
Trigonometry . Vol. Part I: Plane Trigonometry. New York:
Henry Holt and Company . pp. 11–12.
^
a
b Aufmann, Richard; Nation, Richard (2014).
Algebra and Trigonometry (8 ed.).
Cengage Learning . p. 528.
ISBN
978-128596583-3 . Retrieved 2017-07-28 .
^
a
b
c
d
e
f
g
h Bales, John W. (2012) [2001].
"5.1 The Elementary Identities" . Precalculus . Archived from
the original on 2017-07-30. Retrieved 2017-07-30 .
^
a
b
c
d
e
Gunter, Edmund (1620). Canon triangulorum .
^
a
b
c
d
e Roegel, Denis, ed. (2010-12-06).
"A reconstruction of Gunter's Canon triangulorum (1620)" (Research report). HAL. inria-00543938.
Archived from the original on 2017-07-28. Retrieved 2017-07-28 .
^
Weisstein, Eric Wolfgang .
"Coversine" .
MathWorld .
Wolfram Research, Inc.
Archived from the original on 2005-11-27. Retrieved 2015-11-06 .
^
Weisstein, Eric Wolfgang .
"Covercosine" .
MathWorld .
Wolfram Research, Inc.
Archived from the original on 2014-03-28. Retrieved 2015-11-06 .