Cofunction

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.^[1]

For example, sine and cosine 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 and cosecant and of tangent and cotangent:

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

These equations are also known as the cofunction identities.^[1]

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):

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

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

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

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

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

See also

• Vercosine (versed cosine)
• Havercosine (half-versed cosine)
• Covariance

References

1. Aufmann, Richard; Nation, Richard (2014), Algebra and Trigonometry (8th ed.), Cengage Learning, p. 528, ISBN 9781285965833.