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Gauss's Formulas

Gauss's formulas relate the sides and opposite angles of a spherical triangle. Let a spherical triangle have sides a, b, and c with A, B, and C the corresponding opposite angles. Then

(sin[1/2(a-b)])/(sin(1/2c))=(sin[1/2(A-B)])/(cos(1/2C))
(1)
(sin[1/2(a+b)])/(sin(1/2c))=(cos[1/2(A-B)])/(sin(1/2C))
(2)
(cos[1/2(a-b)])/(cos(1/2c))=(sin[1/2(A+B)])/(cos(1/2C))
(3)
(cos[1/2(a+b)])/(cos(1/2c))=(cos[1/2(A+B)])/(sin(1/2C)).
(4)

These formulas are also known as Delambre's analogies (Smart 1960, p. 22).

See also

Spherical Trigonometry

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References

Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, pp. 131 and 147-150, 1987.Smart, W. M. Text-Book on Spherical Astronomy, 6th ed. Cambridge, England: Cambridge University Press, 1960.Zwillinger, D. (Ed.). "Spherical Geometry and Trigonometry." §6.4 in CRC Standard Mathematical Tables and Formulae. Boca Raton, FL: CRC Press, pp. 468-471, 1995.

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Gauss's Formulas

Cite this as:

Weisstein, Eric W. "Gauss's Formulas." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/GausssFormulas.html

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