Spherical law of cosines

In spherical trigonometry, the law of cosines (also called the cosine rule for sides[1]) is a theorem relating the sides and angles of spherical triangles, analogous to the ordinary law of cosines from plane trigonometry.

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:[2][1]

\cos(c) = \cos(a) \cos(b) %2B \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, they are the distances divided by the radius). As a special case, for C = \pi/2 , then \cos(C) =0 \, and one obtains the spherical analogue of the Pythagorean theorem:

\cos(c) = \cos(a) \cos(b). \,

A variation on the law of cosines, the second spherical law of cosines,[3] (also called the cosine rule for angles[1]) states:

\cos(A) = -\cos(B)\cos(C) %2B \sin(B)\sin(C)\cos(a) \,

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.

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.[4]

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 \approx a^2 %2B b^2 - 2ab\cos(C) . \,\!

The error in this approximation, which can be obtained from the Maclaurin series for the cosine and sine functions, is of order

O(c^4) %2B O(a^3 b) %2B O(a b^3) . \,\!

Contents

Proof

A proof of the law of cosines can be constructed as follows.[2] 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) = \mathbf{u} \cdot \mathbf{v}
\cos(b) = \mathbf{u} \cdot \mathbf{w}
\cos(c) = \mathbf{v} \cdot \mathbf{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:

\mathbf{t}_a = \frac{\mathbf{v} - \mathbf{u} (\mathbf{u} \cdot \mathbf{v})}{\left| \mathbf{v} - \mathbf{u} (\mathbf{u} \cdot \mathbf{v}) \right|} = \frac{\mathbf{v} - \mathbf{u} \cos(a)}{\sin(a)}

where for the denominator we have used the Pythagorean identity sin2(a) = 1 − cos2(a). Similarly,

\mathbf{t}_b = \frac{\mathbf{w} - \mathbf{u} \cos(b)}{\sin(b)}.

Then, the angle C is given by:

\cos(C) = \mathbf{t}_a \cdot \mathbf{t}_b = \frac{\cos(c) - \cos(a) \cos(b)}{\sin(a) \sin(b)}

from which the law of cosines immediately follows.

Proof without vectors

To the diagram above, add a plane tangent to the sphere at u, and extend radii from the center of the sphere O 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 sec^2 of any angle is tan^2 %2B 1),

\tan^2 a %2B \tan^2 b - 2\tan a \tan b \cos C \;\;\;=\;\;\; \sec^2 a %2B \sec^2 b - 2 \sec a \sec b \cos c

\;\;= \;\; 2 %2B \tan^2 a %2B \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.

See also

Notes

  1. ^ a b c W. Gellert, S. Gottwald, M. Hellwich, H. Kästner, and H. Küstner, The VNR Concise Encyclopedia of Mathematics, 2nd ed., ch. 12 (Van Nostrand Reinhold: New York, 1989).
  2. ^ a b Romuald Ireneus 'Scibor-Marchocki, Spherical trigonometry, Elementary-Geometry Trigonometry web page (1997).
  3. ^ Reiman, István (1999). Geometria és határterületei. Szalay Könyvkiadó és Kereskedőház Kft.. p. 83. 
  4. ^ R. W. Sinnott, "Virtues of the Haversine", Sky and Telescope 68 (2), 159 (1984).