单词 | Spherical Trigonometry | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
释义 | Spherical TrigonometryDefine a Spherical Triangle on the surface of a unit Sphere, centered at a point
Then
Equivalently,
Since these two expressions are equal, we obtain the identity
The identity
where ![]()
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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,
and the angles of a spherical triangle,
(Beyer 1987), as well as the Law of Tangents
Let
then the half-angle formulas are
where
where
![]() Additional formulas include the Haversine formulas
Gauss's Formulas
and Napier's Analogies
(Beyer 1987).See also Angular Defect, Descartes Total Angular Defect, Gauss's Formulas, Girard's Spherical ExcessFormula, Law of Cosines, Law of Sines, Law of Tangents, L'Huilier's Theorem, Napier'sAnalogies, Spherical Excess, Spherical Geometry, Spherical Polygon, Spherical Triangle 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. |
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