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]^[2] The prefix "co-" can be found already in Edmund Gunter's Canon triangulorum (1620).^[3]^[4]

For example, sine (Latin: sinus) and cosine (Latin: cosinus,^[3]^[4] sinus complementi^[3]^[4]) are cofunctions of each other (hence the "co" in "cosine"):

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

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,^[3]^[4] tangens complementi^[3]^[4]):

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

These equations are also known as the cofunction identities.^[1]^[2]

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):

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

References

1 2 Aufmann, Richard; Nation, Richard (2014). Algebra and Trigonometry (8 ed.). Cengage Learning. p. 528. ISBN 978-128596583-3. Retrieved 2017-07-28.
1 2 3 4 5 6 7 8 Bales, John W. (2012) [2001]. "5.1 The Elementary Identities". Precalculus. Archived from the original on 2017-07-30. Retrieved 2017-07-30.
1 2 3 4 5 Gunter, Edmund (1620). Canon triangulorum.
1 2 3 4 5 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.

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Cofunction

See also

References