| 释义 |
SineLet  be an Angle measured counterclockwise from the x-Axis along the arc of the Unit Circle.Then  is the vertical coordinate of the arc endpoint. As a result of this definition, the sine function isperiodic with period  . By the Pythagorean Theorem, also obeys the identity
 | (1) |
The sine function can be defined algebraically by the infinite sum
 | (2) |
 | (3) |
 | (4) |
 | (5) |
Using the results from the Exponential Sum Formulas
Similarly,
 | (7) |
 | (8) |
 | (9) |
 is a Binomial Coefficient.
Cvijovic and Klinowski (1995) show that the sum
 | (10) |
 ,
 | (11) |
 is an Euler Polynomial.
A Continued Fraction representation of  is
 | (12) |
The value of  is Irrational for all  except 4 and 12, for which  and  .
The Fourier Transform of  is given by
Definite integrals involving include
where  is the Gamma Function.See also Andrew's Sine, Cosecant, Cosine, Fourier Transform--Sine,Hyperbolic Sine, Sinc Function, Tangent, Trigonometry
References
Abramowitz, M. and Stegun, C. 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.Cvijovic, 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. Project Mathematics! Sines and Cosines, Parts I-III. Videotapes (28, 30, and 30 minutes). California Institute of Technology. Available from the Math. Assoc. Amer. 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.
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