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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
| The definition of the cosine function can be extended lớn complex arguments  using the definition
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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,
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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
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(Hardy 1959), where the difference between and Hardy"s approximation is plotted above.
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The cosine obeys the identity
See alsoCis, 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 classroomRelated x-lair.com siteshttp://functions.x-lair.com.com/ElementaryFunctions/Cos/Explore with x-lair.com|Alpha
ReferencesAbramowitz, 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.Referenced on x-lair.com|AlphaCosineCite 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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