The sine function

is one of the basic functions encountered in trigonometry (the others being the cosecant, cosine, cotangent, secant, & tangent). Let
be an angle measured counterclockwise from the x-axis along an arc of the unit circle. Then
is the vertical coordinate of the arc endpoint, as illustrated in the left figure above.

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The common schoolbook definition of the sine of an angle

in a right triangle (which is equivalent to lớn the definition just given) is as the ratio of the lengths of the side of the triangle opposite the angle and the hypotenuse, i.e.,

A convenient mnemonic for remembering the definition of the sine, as well as the 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 sine function is periodic with period

. By the Pythagorean theorem,
also obeys the identity

 Min Max Re Im

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

, illustrated above, using the definition

where e is the base of the natural logarithm và i is the imaginary number. Sine is an entire function và is implemented in the romanhords.com Language as Sin.

A related function known as the hyperbolic sineis similarly defined,

for all

(Calogero 1999; Beylkin và Mohlenkamp 2002; Trott 2005, pp.5-6).

Cvijović and Klinowski (1995) show that the sum

(Olds 1963, p.138).

The value of

is irrational for all integers
1" /> except 2, 4, and 12, for which
,
, và
, respectively, a result that is essentially known as Niven"s theorem.

The Fourier transform of

is given by

for

1" /> where
is the gamma function (R.Mabry, pers. Comm., Dec.15, 2005; T.Drane, pers. Comm., Apr.21, 2006).

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Andrew"s Sine, Cis, Cosecant, Cosine, Elementary Function, Fourier Transform--Sine, Hyperbolic Polar Sine, Hyperbolic Sine, Hypersine, Inverse Sine, Niven"s Theorem, Polar Sine, Sinc Function, Sinusoid, SOHCAHTOA, Tangent, Trigonometric Functions, Trigonometry Explore this topic in the romanhords.com classroom

## Related romanhords.com sites

http://functions.romanhords.com.com/ElementaryFunctions/Sin/

## References

Abramowitz, M. & 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.Beylkin, G. And Mohlenkamp, M.J. Proc. Nat. Acad. Sci. USA 99, 10246, 2002.Beyer, W.H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p.225, 1987.Borwein, J.; Bailey, D.; & Girgensohn, R. Experimentation in Mathematics: Computational Paths lớn Discovery. Wellesley, MA: A K Peters, 2004.Calogero, F. "Remarkable Matrices & Trigonometric Identities. II." Commun. Appl. Math. 3, 267-270, 1999.Cvijović, D. & Klinowski, J. "Closed-Form Summation of Some Trigonometric Series." Math. Comput. 64, 205-210, 1995.Edwards, H.M. Riemann"s Zeta Function. New York: Dover, 2001.Hansen, E.R. A Table of Series and Products. Englewood Cliffs, NJ: Prentice-Hall, 1975.Olds, C.D. Continued Fractions. New York: Random House, 1963.Project Mathematics. "Sines and Cosines, Parts I-III." Videotape. Http://www.projectmathematics.com/sincos1.htm.Jeffrey, A. "Trigonometric Identities." §2.4 in Handbook of Mathematical Formulas và Integrals, 2nd ed. Orlando, FL: Academic Press, pp.111-117, 2000.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 & 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 & Formulae. Boca Raton, FL: CRC Press, pp.452-460, 1995.

Sine

## Cite this as:

Weisstein, Eric W. "Sine." From romanhords.com--Aromanhords.com website Resource. Https://romanhords.com/Sine.html

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