Spherical Triangles

A spherical triangle is the figure made by connecting 3 points on the surface of a sphere (not all on a great circle) with arcs of great circles. Great circles are the intersection of the plane containing the center of the sphere with the surface of the sphere itself. Examples of great circles on the earth are the equator and meridians of longitude.

The parts of a spherical triangle are called sides and angles. Sides are measured by their angles subtended at the center of the sphere. In the figures below the sides are labeled a, b, and c. A, B, and C are the angles opposite sides a, b, and c respectively.

A useful special case is Right Angled Spherical Triangles, where one of the angles is 90 degrees. The equations simplify for this case.

General Spherical Triangles

The Law of Sines

sin a/sin A = sin b/sin B = sin c/sin C

The Law of Cosines

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

Similar relations hold for the other sides and angles.

Right Angled Spherical Triangles

Napier's Rules for a Right Angled Spherical Triangle

Excluding the right angle C arrange the remaining five parts of the spherical right triangle as shown in the circle. Label the three parts opposite right angle C with the prefix co- meaning compliment (90-angle).

Any one part of the circle is called a middle part, the two neighboring parts are called adjacent parts, and the two remaining parts are called opposite parts.

Napier's Rules are:

The sine of any middle part equals the product of the tangents of the adjacent parts.

The sine of any middle part equals the product of the cosines of the opposite parts. Example:

    co-A = 90-A, co-B = 90-B

    sin a = tan b tab (co-B)        or      sin a = tan b cot B
    sin (co-A) = cos a cos (co-B)   or      cos A = cos a sin B