Law of cosines (spherical)

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In spherical trigonometry, the law of cosines (also called the cosine rule for sides) refers to 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 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).

If the law of cosines is used to solve for c, it is susceptible to 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 = 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) . \,\!]

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 \cdot \mathbf]

[\cos(b) = \mathbf \cdot \mathbf]

[\cos(c) = \mathbf \cdot \mathbf]

To get the angle C, we need the tangent vectors t_a and t_b at u along the directions of sides a and b, respectively. For example, the tangent vector t_a 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_a = \frac - \mathbf (\mathbf \cdot \mathbf)} - \mathbf (\mathbf \cdot \mathbf) \right|} = \frac - \mathbf \cos(a)}]

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

[\mathbf_b = \frac - \mathbf \cos(b)}]

Then, the angle C is given by:

[\cos(C) = \mathbf_a \cdot \mathbf_b = \frac]

from which the law of cosines immediately follows.

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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