B2 - - - THE RIGHT SPHERICAL TRIANGLE - - - 10/03/2004

A right spherical triangle is one which has an angle equal to
90 degrees. A spherical triangle unlike a plane triangle, may
have two or even three right angles.

If a triangle has three right angles, we have the solution at
once, for each of the sides is a quadrant or 90 degrees.

If a triangle has two right angles, the sides opposite these angles
are quadrants, and the third angle is measured by its opposite side. If
the third angle or its opposite side is given, the solution is
obvious.

Hence we have to consider the solution of right spherical
triangles having only one right angle.

Let a, b, and c be the sides of a right triangle with opposite
angles A, B, and C. ( C = 90 degrees )

sin a = sin A * sin c sin a = tan b * ctn B
sin b = sin B * sin c sin b = tan a * ctn A
cos c = cos a * cos b cos c = ctn A * ctn B
cos A = cos a * sin B cos A = tan b * ctn c
cos B = cos b * sin A cos B = tan a * ctn c

NAPIERS RULES

The preceding ten formulas can be easily obtained from two
simple rules discovered by John Napier (1550-1617), the
inventor of logarithms. Since the right angle does not enter
into the formulas, only five parts are considered. These are
a, b, and the complements of A, B, and c (or 90-A, 90-B, 90-c)
which can be written A', B', and c'.

If these five parts are arranged in the
c' order in which they occur in the triangle,
any part may be selected and called the
A' B' middle part; then the two parts next to it
* are called adjacent parts, and the other two
are called opposite parts. Whenever any three
b a parts are considered it is always possible to
select one of them in such a manner that the
other two parts will be either adjacent to it
or opposite to it.
Napiers rules are as follows:
1. The sine of the middle part equals the product of the
tangents of the adjacent parts.
2. The sine of the middle part equals the product of the
cosines of the opposite parts.