Let be an Angle measured counterclockwise from the -axis along the arc of the unit Circle. Then is the horizontal coordinate of the arc endpoint. As a result of this definition, the cosine function is periodic with period .
The cosine function has a Fixed Point at 0.739085.
The cosine function can be defined algebraically using the infinite sum
(1) |
(2) |
(3) |
The Fourier Transform of
is given by
(4) |
The cosine sum rule gives an expansion of the Cosine function of a multiple Angle in terms of a sum of Powers of sines and cosines,
(5) |
(6) |
(7) |
(8) |
(9) |
Cvijovic and Klinowski (1995) note that the following series
(10) |
(11) |
See also Euler Polynomial, Exponential Sum Formulas, Fourier Transform--Cosine, Hyperbolic Cosine, Sine, Tangent, Trigonometric Functions
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.
Hardy, G. H. Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work, 3rd ed. New York: Chelsea, p. 68,
1959.
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