Geometry > Trigonometry > Spherical Trigonometry >

Napier's Analogies

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

(Smart 1960, p. 23).

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