The cosine function
is one of the basic functions encountered in trigonometry
(the others being the cosecant , cotangent ,
secant , sine , and tangent ).
Let be an angle
measured counterclockwise from the x -axis along
the arc of the unit circle .
Then is the horizontal coordinate
of the arc endpoint.

The common schoolbook definition of the cosine of an angle in a right triangle
(which is equivalent to the definition just given) is as the ratio of the lengths
of the side of the triangle adjacent to the angle and the hypotenuse ,
i.e.,

(1)

A convenient mnemonic for remembering the definition of the sine , cosine, and tangent is SOHCAHTOA
(sine equals opposite over hypotenuse, cosine equals adjacent
over hypotenuse, tangent equals opposite over adjacent).

As a result of its definition, the cosine function is periodic with period . By the Pythagorean theorem ,
also obeys the identity

(2)

The definition of the cosine function can be extended to complex arguments using the definition

(3)

where e is the base of the natural logarithm and i is the imaginary
number . Cosine is an entire function and is
implemented in the Wolfram Language
as Cos [z ].

A related function known as the hyperbolic cosine
is similarly defined,

(4)

The cosine function has a fixed point at 0.739085... (OEIS A003957 ), a value sometimes known as
the Dottie number (Kaplan 2007).

The cosine function can be defined analytically using the infinite sum

or the infinite product

(7)

A close approximation to
for is

(Hardy 1959), where the difference between and Hardy's approximation is plotted above.

The cosine obeys the identity

(10)

and the multiple-angle formula

(11)

where is a binomial
coefficient . It is related to via

(12)

(Trott 2006, p. 39).

Summation of
from to can be done in closed form as

Similarly,

(18)

where . The exponential
sum formula gives

The sum of
can also be done in closed form,

(21)

The Fourier transform of is given by

where is the delta
function .

Cvijović and Klinowski (1995) note that the following series

(24)

has closed form for ,

(25)

where is an Euler
polynomial .

A definite integral involving is given by

(26)

for where is the gamma function
(T. Drane, pers. comm., Apr. 21, 2006).

See also Cis ,

Dottie Number ,

Elementary Function ,

Euler
Polynomial ,

Exponential Sum Formulas ,

Fourier Transform--Cosine ,

Hyperbolic
Cosine ,

Inverse Cosine ,

Secant ,

Sine ,

SOHCAHTOA ,

Tangent ,

Trigonometric Functions ,

Trigonometry Explore this topic in the
MathWorld classroom
Related Wolfram sites http://functions.wolfram.com/ElementaryFunctions/Cos/
Explore with Wolfram|Alpha
References Abramowitz, M. and Stegun, I. A. (Eds.). "Circular Functions." §4.3 in Handbook
of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing.
New York: Dover, pp. 71-79, 1972. Beyer, W. H. CRC
Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 215,
1987. Cvijović, D. and Klinowski, J. "Closed-Form Summation
of Some Trigonometric Series." Math. Comput. 64 , 205-210, 1995. Hansen,
E. R. A
Table of Series and Products. Englewood Cliffs, NJ: Prentice-Hall, 1975. Hardy,
G. H. Ramanujan:
Twelve Lectures on Subjects Suggested by His Life and Work, 3rd ed. New York:
Chelsea, p. 68, 1959. Jeffrey, A. "Trigonometric Identities."
§2.4 in Handbook
of Mathematical Formulas and Integrals, 2nd ed. Orlando, FL: Academic Press,
pp. 111-117, 2000. Kaplan, S. R. "The Dottie Number."
Math. Mag. 80 , 73-74, 2007. Project Mathematics .
"Sines and Cosines, Parts I-III." Videotape. http://www.projectmathematics.com/sincos1.htm .Sloane,
N. J. A. Sequence A003957 in "The
On-Line Encyclopedia of Integer Sequences." Spanier, J. and Oldham,
K. B. "The Sine
and Cosine
Functions." Ch. 32 in An
Atlas of Functions. Washington, DC: Hemisphere, pp. 295-310, 1987. Tropfke,
J. Teil IB, §1. "Die Begriffe des Sinus und Kosinus eines Winkels."
In Geschichte der Elementar-Mathematik in systematischer Darstellung mit besonderer
Berücksichtigung der Fachwörter, fünfter Band, zweite aufl. Berlin
and Leipzig, Germany: de Gruyter, pp. 11-23, 1923. Trott, M. The
Mathematica GuideBook for Symbolics. New York: Springer-Verlag, 2006. http://www.mathematicaguidebooks.org/ . Zwillinger,
D. (Ed.). "Trigonometric or Circular Functions." §6.1 in CRC
Standard Mathematical Tables and Formulae. Boca Raton, FL: CRC Press, pp. 452-460,
1995. Referenced on Wolfram|Alpha Cosine
Cite this as:
Weisstein, Eric W. "Cosine." From MathWorld --A
Wolfram Web Resource. https://mathworld.wolfram.com/Cosine.html

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