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) |

Similarly,

(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) |

where is the Gamma Function.

**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

1999-05-26