gifSpherical Trigonometry

This page gives a basic introduction to spherical trigonometry with instructions on how to build a model that relates a spherical triangle to four planar triangles that form the sides of a pyramid in which it lies. The model is used to derive the cosine rule for sides of a spherical triangle.

Recall the cosine rule for sides of a planar triangle which gives the third side when the other two sides and the angle between them are known:

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a2 = b2 + c2 - 2bc cos α

b2 = a2 + c2 - 2ac cos β

c2 = a2 + b2 - 2ab cos γ

It is desirable to find the analagous formulae for spherical triangles...

Cosine Rule for Sides of a Spherical Triangle

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The figure above shows a spherical triangle ABC formed by the intersection of three great circles with their origin O at the center of a unit sphere. The edges of the flat planes in which the arcs a, b, c lie, meet along OA, OBQ, and OCP. The edge AQ is tangential to the arc c, and the edge AP is tangential to the arc b. Therefore OAQ and OAP are right angles.

The edges of the three planes in which the arcs a, b, c lie forms the triangle PAQ of which the apex angle PAQ is equivalent to the angle α of the spherical triangle. In other words, the spherical angle α is the angle between the planes OAQ and OAP.

It is a simple matter to build a small model that shows the relation of the spherical triangle to the parts of four planar triangles that form the sides of a pyramid in which it lies. The model is best made from acetate sheet, inscribed with permanent markers. The plan is given below. If you have a flat screen monitor, you can lay an acetate sheet over the screen and carefully mark off the vertices with a permanent pen. (Do not apply pressure to the sceen or you will damage the LCD).

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Examing the unfolded model, and using the cosine rule for planar triangles, it is easy to see that:

   PQ2 = PO2 + QO2 - 2PO.QO cos a

   PQ2 = PA2 + QA2 - 2PA.QA cos α

Therefore:

   (PO2 - PA2) + (QO2 - QA2) - 2PO.QO cos a + 2PA.QA cos α = 0

   2PO.QO cos a = 2AO2 + 2PA.QA cos α

Divide through by 2PO.QO, then:

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            = cos POA cos QOA + sin POA sin QOA cos α

            = cos b cos c + sin b sin c cos α

Thus the fomula for solving a spherical triangle which gives the third side (a) when the other two sides (b and c) and the angle between them (α) are known is:

   cos a = cos b cos c + sin b sin c cos α

Model disection

Studying each planar triangle of the unfolded model, we have:

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If we make other models with tangential right-triangles at β and γ respectively, we can solve b and c in the same manner. Thus the cosine rules for solving the sides of a spherical triangle are:

   cos a = cos b cos c + sin b sin c cos α

   cos b = cos c cos a + sin c sin a cos β

   cos c = cos a cos b + sin a sin b cos γ

Cosine Rule for Angles of a Spherical Triangle

The cosine rules for the angles of a spherical triangle are obtained by cyclic permutation (when the sides and angles are less than π):

   cos α = -cos β cos γ + sin β sin γ cos a

   cos β = -cos γ cos α + sin γ sin α cos b

   cos γ = -cos α cos β + sin α sin β cos c


[Refs: ‘The VNR Concise Encyclopedia of Mathematics’ pp.262 - 265. W. Gellert, H. Küstner, M. Hellwich, H.Kästner (K.A Hirsch, H. Reichart), Bibliographisches, Institut Leipzig 1975; ‘Mathematics for The Million’ pp. 368-369. L. Hogben (George Allen & Unwin Ltd., 1936).]