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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.
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