Cofunction information

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"):

$\sin \left({\frac {\pi }{2}}-A\right)=\cos(A)$ ^[1]^[3]	$\cos \left({\frac {\pi }{2}}-A\right)=\sin(A)$ ^[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]):

$\sec \left({\frac {\pi }{2}}-A\right)=\csc(A)$ ^[1]^[3]	$\csc \left({\frac {\pi }{2}}-A\right)=\sec(A)$ ^[1]^[3]
$\tan \left({\frac {\pi }{2}}-A\right)=\cot(A)$ ^[1]^[3]	$\cot \left({\frac {\pi }{2}}-A\right)=\tan(A)$ ^[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):

$\operatorname {ver} \left({\frac {\pi }{2}}-A\right)=\operatorname {cvs} (A)$ ^[6]	$\operatorname {cvs} \left({\frac {\pi }{2}}-A\right)=\operatorname {ver} (A)$
$\operatorname {vcs} \left({\frac {\pi }{2}}-A\right)=\operatorname {cvc} (A)$ ^[7]	$\operatorname {cvc} \left({\frac {\pi }{2}}-A\right)=\operatorname {vcs} (A)$
$\operatorname {hav} \left({\frac {\pi }{2}}-A\right)=\operatorname {hcv} (A)$	$\operatorname {hcv} \left({\frac {\pi }{2}}-A\right)=\operatorname {hav} (A)$
$\operatorname {hvc} \left({\frac {\pi }{2}}-A\right)=\operatorname {hcc} (A)$	$\operatorname {hcc} \left({\frac {\pi }{2}}-A\right)=\operatorname {hvc} (A)$
$\operatorname {exs} \left({\frac {\pi }{2}}-A\right)=\operatorname {exc} (A)$	$\operatorname {exc} \left({\frac {\pi }{2}}-A\right)=\operatorname {exs} (A)$

^ ^a ^b ^c ^d ^e ^f ^g Cite error: The named reference Hall_1909 was invoked but never defined (see the help page).
^ ^a ^b Cite error: The named reference Aufmann_Nation_2014 was invoked but never defined (see the help page).
^ ^a ^b ^c ^d ^e ^f ^g ^h Cite error: The named reference Bales_2012 was invoked but never defined (see the help page).
^ ^a ^b ^c ^d ^e Cite error: The named reference Gunter_1620 was invoked but never defined (see the help page).
^ ^a ^b ^c ^d ^e Cite error: The named reference Roegel_2010 was invoked but never defined (see the help page).
^ Cite error: The named reference Weisstein_covers was invoked but never defined (see the help page).
^ Cite error: The named reference Weisstein_covercos was invoked but never defined (see the help page).

[Hall_1909-1] ^ ^a ^b ^c ^d ^e ^f ^g Cite error: The named reference Hall_1909 was invoked but never defined (see the help page).

[Aufmann_Nation_2014-2] Cite error: The named reference Aufmann_Nation_2014 was invoked but never defined (see the help page).

[Bales_2012-3] ^ ^a ^b ^c ^d ^e ^f ^g ^h Cite error: The named reference Bales_2012 was invoked but never defined (see the help page).

[Gunter_1620-4] Cite error: The named reference Gunter_1620 was invoked but never defined (see the help page).

[Roegel_2010-5] Cite error: The named reference Roegel_2010 was invoked but never defined (see the help page).

[Weisstein_covers-6] Cite error: The named reference Weisstein_covers was invoked but never defined (see the help page).

[Weisstein_covercos-7] Cite error: The named reference Weisstein_covercos was invoked but never defined (see the help page).

Cofunction information

and 8 Related for: Cofunction information

Cofunction

CO

Inverse trigonometric functions

Glossary of calculus

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Outline of trigonometry

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Hermann Schapira