Define a Spherical Triangle on the surface of a unit Sphere, centered at a point , with vertices , ,
and . Define Angles
,
, and
. Let the
Angle between Planes and be , the Angle between Planes
and be , and the Angle between Planes and be . Define the
Vectors

(1) | |||

(2) | |||

(3) |

Then

(4) |

Equivalently,

(5) |

Since these two expressions are equal, we obtain the identity

(6) |

The identity

(7) |

where is the Scalar Triple Product, gives a spherical analog of the Law of Sines,

(8) |

(9) | |||

(10) |

These are the fundamental equalities of spherical trigonometry.

There are also spherical analogs of the Law of Cosines for the sides of a spherical triangle,

(11) | |||

(12) | |||

(13) |

and the angles of a spherical triangle,

(14) | |||

(15) | |||

(16) |

(Beyer 1987), as well as the Law of Tangents

(17) |

Let

(18) | |||

(19) |

then the half-angle formulas are

(20) | |||

(21) | |||

(22) |

where

(23) |

(24) | |||

(25) | |||

(26) |

where

(27) |

Additional formulas include the Haversine formulas

(28) | |||

(29) | |||

(30) | |||

(31) |

Gauss's Formulas

(32) | |||

(33) | |||

(34) | |||

(35) |

and Napier's Analogies

(36) | |||

(37) | |||

(38) | |||

(39) |

(Beyer 1987).

**References**

Beyer, W. H. *CRC Standard Mathematical Tables, 28th ed.* Boca Raton, FL: CRC Press, pp. 131 and 147-150, 1987.

Danby, J. M. *Fundamentals of Celestial Mechanics, 2nd ed., rev. ed.* Richmond, VA: Willmann-Bell, 1988.

Smart, W. M. *Text-Book on Spherical Astronomy, 6th ed.* Cambridge, England: Cambridge University Press, 1960.

© 1996-9

1999-05-26