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.

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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 và the hypotenuse, i.e.,

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

 Min Max Re Im

The definition of the cosine function can be extended lớn complex arguments

using the definition

where e is the base of the natural logarithm & i is the imaginary number. Cosine is an entire function and is implemented in the x-lair.com Language as Cos.

A related function known as the hyperbolic cosineis similarly defined,

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

(Hardy 1959), where the difference between

and Hardy"s approximation is plotted above.

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The cosine obeys the identity

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 x-lair.com classroom

## Related x-lair.com sites

http://functions.x-lair.com.com/ElementaryFunctions/Cos/

## References

Abramowitz, M. Và Stegun, I.A. (Eds.). "Circular Functions." §4.3 in Handbook of Mathematical Functions with Formulas, Graphs, & 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. & Klinowski, J. "Closed-Form Summation of Some Trigonometric Series." Math. Comput. 64, 205-210, 1995.Hansen, E.R. A Table of Series & 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 & 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 & 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. Và Oldham, K.B. "The Sine
và 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.

Cosine

## Cite this as:

Weisstein, Eric W. "Cosine." From x-lair.com--Ax-lair.com web Resource. Https://x-lair.com/Cosine.html

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