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) |

where is the Delta Function.

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) |

Summing the Cosine of a multiple angle from to can be done in closed form using

(6) |

(7) |

Similarly,

(8) |

(9) |

Cvijovic and Klinowski (1995) note that the following series

(10) |

(11) |

**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

1999-05-25