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Spherical law of cosines

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Spherical law of cosines

In spherical trigonometry, the law of cosines (also called the cosine rule for sides) is a theorem relating the sides and angles of spherical triangles, analogous to the ordinary law of cosines from plane trigonometry.

Contents

Given a unit sphere, a "spherical triangle" on the surface of the sphere is defined by the great circles connecting three points u, v, and w on the sphere (shown at right). If the lengths of these three sides are a (from u to v), b (from u to w), and c (from v to w), and the angle of the corner opposite c is C, then the (first) spherical law of cosines states:

cos c = cos a cos b + sin a sin b cos C .

Since this is a unit sphere, the lengths a, b, and c are simply equal to the angles (in radians) subtended by those sides from the center of the sphere. (For a non-unit sphere, the lengths are the subtended angles times the radius, and the formula still holds if a, b and c are reinterpreted as the subtended angles). As a special case, for C = π/2, then cos C = 0, and one obtains the spherical analogue of the Pythagorean theorem:

cos c = cos a cos b .

If the law of cosines is used to solve for c, the necessity of inverting the cosine magnifies rounding errors when c is small. In this case, the alternative formulation of the law of haversines is preferable.

A variation on the law of cosines, the second spherical law of cosines, (also called the cosine rule for angles) states:

cos C = cos A cos B + sin A sin B cos c ,

where A and B are the angles of the corners opposite to sides a and b, respectively. It can be obtained from consideration of a spherical triangle dual to the given one.

First proof

A proof of the law of cosines can be constructed as follows. Let u, v, and w denote the unit vectors from the center of the sphere to those corners of the triangle. Then, the lengths (angles) of the sides are given by the dot products:

cos a = u v cos b = u w cos c = v w

To get the angle C, we need the tangent vectors ta and tb at u along the directions of sides a and b, respectively. For example, the tangent vector ta is the unit vector perpendicular to u in the u-v plane, whose direction is given by the component of v perpendicular to u. This means:

t a = v u ( u v ) v u ( u v ) = v u cos a sin a

where for the denominator we have used the Pythagorean identity sin2 a = 1 − cos2 a and where ||   || denotes the length of the vector in the denominator. Similarly,

t b = w u cos b sin b .

Then, the angle C is given by:

cos C = t a t b = cos c cos a cos b sin a sin b

from which the law of cosines immediately follows.

Second proof

To the diagram above, add a plane tangent to the sphere at u, and extend radii from the center of the sphere O through v and through w to meet the plane at points y and z. We then have two plane triangles with a side in common: the triangle containing u, y and z and the one containing O, y and z. Sides of the first triangle are tan a and tan b, with angle C between them; sides of the second triangle are sec a and sec b, with angle c between them. By the law of cosines for plane triangles (and remembering that sec2 of any angle is tan2 + 1),

tan 2 a + tan 2 b 2 tan a tan b cos C = sec 2 a + sec 2 b 2 sec a sec b cos c = 2 + tan 2 a + tan 2 b 2 sec a sec b cos c .

So

tan a tan b cos C = 1 sec a sec b cos c

Multiply both sides by cos a cos b and rearrange.

Third proof

The angles and distances do not change if the sphere is rotated, so we can rotate the sphere so that u is at the north pole and v is somewhere on the prime meridian (longitude of 0). With this rotation, the spherical coordinates for v are ( r , θ , ϕ ) = ( 1 , a , 0 ) and the spherical coordinates for w are ( r , θ , ϕ ) = ( 1 , b , C ) . The Cartesian coordinates for v are ( x , y , z ) = ( sin a , 0 , cos a ) and the Cartesian coordinates for w are ( x , y , z ) = ( sin b cos C , sin b sin C , cos b ) . The value of cos c is the dot product of the two Cartesian vectors, which is sin a sin b cos C + cos a cos b .

Fourth proof

The vectors u × v and u × w have lengths sin a and sin b respectively and the angle between them is C, so

sin a sin b cos C = ( u × v ) ( u × w ) = ( u u ) ( v w ) ( u v ) ( u w ) = cos c cos a cos b ,

using u · u = 1, u · v = cos a, u · w = cos b, v · w = cos c, cross products, and the Binet–Cauchy identity (p × q) · (r × s) = (p · r)(q · s) − (p · s)(q · r).

Rearrangements

The first and second spherical laws of cosines can be rearranged to put the sides (a, b, c) and angles (A, B, C) on opposite sides of the equations:

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

Planar limit: small angles

For small spherical triangles, i.e. for small a, b, and c, the spherical law of cosines is approximately the same as the ordinary planar law of cosines,

c 2 a 2 + b 2 2 a b cos C .

To prove this, we will use the small-angle approximation obtained from the Maclaurin series for the cosine and sine functions:

cos a = 1 a 2 2 + O ( a 4 ) , sin a = a + O ( a 3 )

Substituting these expressions into the spherical law of cosines nets:

1 c 2 2 + O ( c 4 ) = 1 a 2 2 b 2 2 + a 2 b 2 4 + O ( a 4 ) + O ( b 4 ) + cos ( C ) ( a b + O ( a 3 b ) + O ( a b 3 ) + O ( a 3 b 3 ) )

or after simplifying:

c 2 = a 2 + b 2 2 a b cos C + O ( c 4 ) + O ( a 4 ) + O ( b 4 ) + O ( a 2 b 2 ) + O ( a 3 b ) + O ( a b 3 ) + O ( a 3 b 3 ) .

Remembering the properties of big O notation, we can discard summands where the lowest exponent for a or b is greater than 1, so finally, the error in this approximation is:

O ( c 4 ) + O ( a 3 b ) + O ( a b 3 ) .

References

Spherical law of cosines Wikipedia