Cofunction

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. This definition typically applies to trigonometric functions. The prefix "co-" can be found already in Edmund Gunter's Canon triangulorum (1620).

For example, sine (Latin: sinus) and cosine (Latin: cosinus, sinus complementi) are cofunctions of each other (hence the "co" in "cosine"):

$\sin \left({\frac {\pi }{2}}-A\right)=\cos(A)$

$\cos \left({\frac {\pi }{2}}-A\right)=\sin(A)$

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, tangens complementi):

$\sec \left({\frac {\pi }{2}}-A\right)=\csc(A)$	$\csc \left({\frac {\pi }{2}}-A\right)=\sec(A)$
$\tan \left({\frac {\pi }{2}}-A\right)=\cot(A)$	$\cot \left({\frac {\pi }{2}}-A\right)=\tan(A)$

These equations are also known as the cofunction identities.

This also holds true for the coversine (coversed sine, cvs), covercosine (coversed cosine, cvc), hacoversine (half-coversed sine, hcv), hacovercosine (half-coversed cosine, hcc) and excosecant (exterior cosecant, exc):

$\operatorname {cvs} \left({\frac {\pi }{2}}-A\right)=\operatorname {ver} (A)$

$\operatorname {cvc} \left({\frac {\pi }{2}}-A\right)=\operatorname {vcs} (A)$

$\operatorname {hcv} \left({\frac {\pi }{2}}-A\right)=\operatorname {hav} (A)$

$\operatorname {hcc} \left({\frac {\pi }{2}}-A\right)=\operatorname {hvc} (A)$

$\operatorname {exc} \left({\frac {\pi }{2}}-A\right)=\operatorname {exs} (A)$

See also

• Vercosine (versed cosine)
• Havercosine (half-versed cosine)
• Hyperbolic cosine
• Hyperbolic cosecant
• Hyperbolic cotangent
• Lemniscatic cosine
• Jacobi elliptic cosine
• Covariance
• List of trigonometric identities