Let , , and be the lengths of the legs of a Triangle opposite Angles , , and .
Then the law of cosines states

(1) |

(2) | |||

(3) | |||

(4) |

where is the Angle between and .

The formula can also be derived using a little geometry and simple algebra. From the above diagram,

(5) |

The law of cosines for the sides of a Spherical Triangle states that

(6) | |||

(7) | |||

(8) |

(Beyer 1987). The law of cosines for the angles of a Spherical Triangle states that

(9) | |||

(10) | |||

(11) |

(Beyer 1987).

**References**

Abramowitz, M. and Stegun, C. A. (Eds.).
*Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing.*
New York: Dover, p. 79, 1972.

Beyer, W. H. *CRC Standard Mathematical Tables, 28th ed.* Boca Raton, FL: CRC Press, pp. 148-149, 1987.

© 1996-9

1999-05-26