Let 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 is
periodic 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
(6) |
(7) |
(8) |
(9) |
Cvijovic and Klinowski (1995) show that the sum
(10) |
(11) |
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
(13) |
Definite integrals involving include
(14) | |||
(15) | |||
(16) | |||
(17) |
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.
© 1996-9 Eric W. Weisstein