Reference Ellipsoid and the Geodetic Coordinate System - Geodesy: Introduction to Geodetic Datum and Geodetic Systems - page 166

Fig. 5.3 Napier's rules for
right-angled spherical
triangles

The law of tangents is:

tan 2 A tan 2

ð þ B Þ ð a þ b Þ

Þ ﾼ Þ :

ð 5 : 8 Þ

tan 2 A tan 2 ð B ð a b

Napier's Rules for Right-Angled Spherical Triangles

Let one angle of a spherical triangle ABC be 90 , then the cosine of this right angle
is 0 and the sine is 1. Substituting into the above formulae, one can obtain the
relations between sides and angles of a right-angled spherical triangle. To facilitate
memorization, Napier presented some rules.
Except for the right angle C, there are five elements of a spherical triangle ABC
arranged in the form of a circle. Keep the two elements adjacent to the right angle
C and replace all elements non-adjacent to the right angle C by their complement to
90 (hypotenuse c and angles A and B; see Fig. 5.3 ). Then Napier's rules hold that
the sine of any element in the circle is equal to:
1. The product of the tangents of the adjacent two elements
2. The product of the cosines of the opposite two elements
Take the angle 90 c for example; its adjacent elements are 90 A and

90

B, and its opposite elements are a and b. Hence:

sin 90 tan 90 tan 90

ð c Þﾼ ð A Þ ð B Þ and

sin 90 ð c Þﾼ cos a cos b :

Namely:

cos c ﾼ cot A cot B, and

cos c ﾼ cos a cos b :