Law of cosines (spherical)

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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.

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, they are the distances divided by the radius).

Spherical triangle solved by the law of cosines.
Spherical triangle solved by the law of cosines.

The second spherical law of cosines states:

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

It can be obtained from consideration of the 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.

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 + b^2 - 2ab\cos(C) . \,\!

The error in this approximation is of order

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

If C = π / 2, then \cos(C) =0 \,, and the spherical Pythagorean theorem reads

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

[edit] 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) = \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.

[edit] References

  • Romuald Ireneus 'Scibor-Marchocki, Spherical trigonometry, Elementary-Geometry Trigonometry web page (1997).
  • 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).
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