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

References

1 2 3 4 5 6 7 Hall, Arthur Graham; Frink, Fred Goodrich (January 1909). "Chapter II. The Acute Angle [10] Functions of complementary angles". Written at Ann Arbor, Michigan, USA. Trigonometry. Part I: Plane Trigonometry. New York, USA: Henry Holt and Company / Norwood Press / J. S. Cushing Co. - Berwick & Smith Co., Norwood, Massachusetts, USA. pp. 11–12. Retrieved 2017-08-12.
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.
↑ 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.

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[Hall_1909-1] 1 2 3 4 5 6 7 Hall, Arthur Graham; Frink, Fred Goodrich (January 1909). "Chapter II. The Acute Angle [10] Functions of complementary angles". Written at Ann Arbor, Michigan, USA. Trigonometry. Part I: Plane Trigonometry. New York, USA: Henry Holt and Company / Norwood Press / J. S. Cushing Co. - Berwick & Smith Co., Norwood, Massachusetts, USA. pp. 11–12. Retrieved 2017-08-12.

[Aufmann_Nation_2014-2] 1 2 Aufmann, Richard; Nation, Richard (2014). Algebra and Trigonometry (8 ed.). Cengage Learning. p. 528. ISBN 978-128596583-3. Retrieved 2017-07-28.

[Bales_2012-3] 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.

[Gunter_1620-4] 1 2 3 4 5 Gunter, Edmund (1620). Canon triangulorum.

[Roegel_2010-5] 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.

[Weisstein_covers-6] Weisstein, Eric Wolfgang. "Coversine". MathWorld. Wolfram Research, Inc. Archived from the original on 2005-11-27. Retrieved 2015-11-06.

[Weisstein_covercos-7] Weisstein, Eric Wolfgang. "Covercosine". MathWorld. Wolfram Research, Inc. Archived from the original on 2014-03-28. Retrieved 2015-11-06.

Cofunction

See also

References