Spherical law of cosines

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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.

Spherical triangle solved by the law of cosines.

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) + \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 = π2, 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) + \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]

Proof[edit]

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) and where || || denotes the length of the vector in the denominator. 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[edit]

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),


\begin{align}
\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 \\[4pt]
& = 2 + \tan^2 a + \tan^2 b - 2 \sec a \sec b \cos c.
\end{align}

So

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

Multiply both sides by cos a cos b and rearrange.

Planar limit: small angles[edit]

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) . \,\!

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

\cos(a) = 1 - \frac{a^2}{2} + O(a^4),\, \sin(a) = a + O(a^3)

Substituting these expressions into the spherical law of cosines nets:

1 - \frac{c^2}{2} + O(c^4) = 1 - \frac{a^2}{2} - \frac{b^2}{2} + \frac{a^2 b^2}{4} + O(a^4) + O(b^4) + \cos(C)(ab + O(a^3 b) + O(ab ^ 3) + O(a^3 b^3))

or after simplifying:

c^2 = a^2 + b^2 - 2ab\cos(C) + O(c^4) + O(a^4) + O(b^4) + O(a^2 b^2) + O(a^3 b) + O(ab ^ 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) . \,\!

See also[edit]

Notes[edit]

  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).