Geometry > Trigonometry > Spherical Trigonometry >

Gauss's Formulas

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).

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