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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
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(1) |
The sine function can be defined algebraically by the infinite sum
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(2) |
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(3) |
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(4) |
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(5) |
Using the results from the Exponential Sum Formulas
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(6) |
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(7) |
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(8) |
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(9) |
Cvijovic and Klinowski (1995) show that the sum
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(10) |
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(11) |
A Continued Fraction representation of is
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(12) |
The value of is Irrational for all
except 4 and 12, for which
and
.
The Fourier Transform of
is given by
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|
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(13) |
Definite integrals involving include
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(14) |
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(15) |
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(16) |
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(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.
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© 1996-9 Eric W. Weisstein