A few useful Spherical Trigonometry Formulae
As in plane trigonometry the angles of a spherical triangle are referred to as A, B and C, and the sides a, b and c are expressed in angular measure.

cos a = cos b x cos c + sin b x sin c x cos A
cos b = cos c x cos a + sin c x sin a x cos B
cos c = cos a x cos b + sin a x sin b x cos C

cos A = -cos B x cos C + sin B x sin C x cos a
cos B = -cos C x cos A + sin C x sin A x cos b
cos C = -cos A x cos B + sin A x sin B x cos c
In any spherical triangle

If three sides of a spherical triangle are known then:

In a right spherical triangle, that is with angle C being a 90° angle the following hold true:

sin a = sin A x sin c
sin b = sin B x sin c

tan a = cos B x tan c
tan a = tan A x sin b
tan b = cos A x tan c
tan b = tan B x sin a

cos A = sin B x cos a
cos B = sin A x cos b
cos c = cos a x cos b
cos c = cot B x cot A

AREA

The area of a spherical triangle as measured on the surface of the earth is:

A + B + C - pi

Where A, B, C are expressed in Radians. To convert degrees to radians multiply by pi/180. The answer will be in Radians squared and needs to be multiplied by (180/pi * 180/pi) to give square degrees which needs to be multiplied by 3600 to give square nautical miles. Alternately multiply the answer by 11818102.86.

The area of any other spherical triangle is:

(A + B + C - pi) x R2

Where R, in radians, is the radius of the sphere on which the triangle occurs.

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Copyright: Eugene Griessel - November 2000