Spherical law of cosines explained

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, and on the sphere (shown at right). If the lengths of these three sides are (from to (from to), and (from to), and the angle of the corner opposite is, then the (first) spherical law of cosines states:^{[1] [2]}

$$\backslash cos\; c\; =\; \backslash cos\; a\; \backslash cos\; b\; +\; \backslash sin\; a\; \backslash sin\; b\; \backslash cos\; C\backslash ,$$

Since this is a unit sphere, the lengths, and 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 and are reinterpreted as the subtended angles). As a special case, for, then, and one obtains the spherical analogue of the Pythagorean theorem:

$$\backslash cos\; c\; =\; \backslash cos\; a\; \backslash cos\; b\backslash ,$$

If the law of cosines is used to solve for, the necessity of inverting the cosine magnifies rounding errors when is small. In this case, the alternative formulation of the law of haversines is preferable.^[3]

A variation on the law of cosines, the second spherical law of cosines,^[4] (also called the cosine rule for angles^[2]) states:

$$\backslash cos\; C\; =\; -\backslash cos\; A\; \backslash cos\; B\; +\; \backslash sin\; A\; \backslash sin\; B\; \backslash cos\; c\backslash ,$$

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

Proofs

First proof

Let, and denote the unit vectors from the center of the sphere to those corners of the triangle. The angles and distances do not change if the coordinate system is rotated, so we can rotate the coordinate system 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, \theta, \phi) = ( 1 , a, 0 ) , where is the angle measured from the north pole not from the equator, and the spherical coordinates for w are (r, \theta, \phi) = ( 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 .

Second proof

Let, and denote the unit vectors from the center of the sphere to those corners of the triangle. We have,,, and . The vectors and have lengths and respectively and the angle between them is, so

using cross products, dot products, and the Binet–Cauchy identity $$(\backslash bold\; p\; \backslash times\; \backslash bold\; q)\; \backslash cdot\; (\backslash bold\; r\; \backslash times\; \backslash bold\; s)\; =\; (\backslash bold\; p\; \backslash cdot\; \backslash bold\; r)(\backslash bold\; q\; \backslash cdot\; \backslash bold\; s)\; -\; (\backslash bold\; p\; \backslash cdot\; \backslash bold\; s)(\backslash bold\; q\; \backslash cdot\; \backslash bold\; r).$$

Third proof

Let, and denote the unit vectors from the center of the sphere to those corners of the triangle. Consider the following rotational sequence where we first rotate the vector to by an angle

a, followed by another rotation of vector to by an angle b, after which we rotate the vector back to by an angle c. These three rotational operations can be represented by quaternions:

where

A , B , and C are the unit vectors representing the axes of rotations, as defined by the right-hand rule, respectively. The composition of these three rotation operations is unity, q _{C} q _{B} q _{A} = 1 . Right multiplying both sides by conjugates q ^{*} _{A} q ^{*} _{B} , we have q _{C} = q ^{*} _{A} q ^{*} _{B} , where $q\_A^*\; =\; \backslash cos\; \backslash frac\; -\; \backslash mathbf\; \backslash sin\; \backslash frac$ and $q\_B^*\; =\; \backslash cos\; \backslash frac\; -\; \backslash mathbf\; \backslash sin\; \backslash frac\; .$ This gives us the identity^{[5] [6]}

$$\backslash cos\; \backslash frac\; +\; \backslash mathbf\; \backslash sin\; \backslash frac\; =\; \backslash left(\backslash cos\; \backslash frac\; -\; \backslash mathbf\; \backslash sin\; \backslash frac\backslash right)\; \backslash left(\backslash cos\; \backslash frac\; -\; \backslash mathbf\; \backslash sin\; \backslash frac\; \backslash right).$$

The quaternion product on the right-hand side of this identity is given by

$$\backslash left(\backslash cos\; \backslash frac\; \backslash cos\; \backslash frac\; -\; \backslash mathbf\; \backslash cdot\; \backslash mathbf\; \backslash sin\; \backslash frac\; \backslash sin\; \backslash frac\; \backslash right)\; -\; \backslash left(\backslash mathbf\; \backslash sin\; \backslash frac\; \backslash cos\; \backslash frac\; +\; \backslash mathbf\; \backslash cos\; \backslash frac\; \backslash sin\; \backslash frac\; -\; \backslash mathbf\; \backslash times\; \backslash mathbf\; \backslash sin\; \backslash frac\; \backslash sin\; \backslash frac\; \backslash right).$$

Equating the scalar parts on both sides of the identity, we have

$$\backslash cos\; \backslash frac\; =\; \backslash cos\; \backslash frac\; \backslash cos\; \backslash frac\; -\; \backslash mathbf\; \backslash cdot\; \backslash mathbf\; \backslash sin\; \backslash frac\; \backslash sin\; \backslash frac.$$

Here

A ⋅ B = \cos (\pi - C) = - \cos C . Since this identity is valid for any arc angles, taking 2 a , 2 b , and 2 c instead of a , b , and c , we recover the cosine law in a more familiar form

$$\backslash cos\; c\; =\; \backslash cos\; a\; \backslash cos\; b\; +\; \backslash cos\; C\; \backslash sin\; a\; \backslash sin\; b.$$

We can also recover the sine law by first noting that

A x B = - u \sin C and then equating the vector parts on both sides of the identity as

$$\backslash mathbf\; \backslash sin\; \backslash frac\; =\; -\backslash left(\backslash mathbf\; \backslash sin\; \backslash frac\; \backslash cos\; \backslash frac\; +\; \backslash mathbf\; \backslash cos\; \backslash frac\; \backslash sin\; \backslash frac\; +\; \backslash mathbf\; \backslash sin\; C\; \backslash sin\; \backslash frac\; \backslash sin\; \backslash frac\; \backslash right).$$

The vector

u is orthogonal to both the vectors A and B , and as such u ⋅ A = u ⋅ B = 0 . Taking dot product with respect to u on both sides, and taking 2 a , 2 b , and 2 c instead of a , b , and c , we have u ⋅ C \sin c = - \sin C \sin a \sin b. Now v x w = - C \sin c and so we have u ⋅ ( v x w ) = - u ⋅ C \sin c = \sin C \sin a \sin b. Dividing each side by \sin a \sin b \sin c , we have

$$\backslash frac\; =\; \backslash frac.$$

Since the right-hand side of the above expression is unchanged by cyclic permutation, we have

$$\backslash frac\; =\; \backslash frac\; =\; \backslash frac.$$

Rearrangements

The first and second spherical laws of cosines can be rearranged to put the sides and angles on opposite sides of the equations:

Planar limit: small angles

For small spherical triangles, i.e. for small, and, the spherical law of cosines is approximately the same as the ordinary planar law of cosines,$$c^2\; \backslash approx\; a^2\; +\; b^2\; -\; 2ab\backslash cos\; C\; \backslash ,.$$

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

Substituting these expressions into the spherical law of cosines nets:

$$1\; -\; \backslash frac\; +\; O\backslash left(c^4\backslash right)\; =\; 1\; -\; \backslash frac\; -\; \backslash frac\; +\; \backslash frac\; +\; O\backslash left(a^4\backslash right)\; +\; O\backslash left(b^4\backslash right)\; +\; \backslash cos(C)\backslash left(ab\; +\; O\backslash left(a^3\; b\backslash right)\; +\; O\backslash left(ab^3\backslash right)\; +\; O\backslash left(a^3\; b^3\backslash right)\backslash right)$$

or after simplifying:

$$c^2\; =\; a^2\; +\; b^2\; -\; 2ab\backslash cos\; C\; +\; O\backslash left(c^4\backslash right)\; +\; O\backslash left(a^4\backslash right)\; +\; O\backslash left(b^4\backslash right)\; +\; O\backslash left(a^2\; b^2\backslash right)\; +\; O\backslash left(a^3\; b\backslash right)\; +\; O\backslash left(ab^3\backslash right)\; +\; O\backslash left(a^3\; b^3\backslash right).$$

The big O terms for and are dominated by as and get small, so we can write this last expression as:

$$c^2\; =\; a^2\; +\; b^2\; -\; 2ab\backslash cos\; C\; +\; O\backslash left(a^4\backslash right)\; +\; O\backslash left(b^4\backslash right)\; +\; O\backslash left(c^4\backslash right).$$

History

Something equivalent to the spherical law of cosines was used (but not stated in general) by al-Khwārizmī (9th century), al-Battānī (9th century), and Nīlakaṇṭha (15th century).^[7]

See also

• Half-side formula
• Hyperbolic law of cosines
• Solution of triangles
• Spherical law of sines

Notes

1. Romuald Ireneus 'Scibor-Marchocki, Spherical trigonometry, Elementary-Geometry Trigonometry web page (1997).
2. 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).
3. R. W. Sinnott, "Virtues of the Haversine", Sky and Telescope 68 (2), 159 (1984).
4. Book: Reiman, István . 1999 . Geometria és határterületei . Szalay Könyvkiadó és Kereskedőház Kft. . 83 .
5. Book: Brand, Louis . Vector and Tensor Analysis . 1947 . Wiley . 416–417 . §186 Great Circle Arccs . https://archive.org/details/vectortensoranal00branrich/page/416/ .
6. Book: Kuipers, Jack B. . Quaternions and Rotation Sequences . 1999 . Princeton University Press . 235–255 . §10 Spherical Trignometry .
7. Book: Van Brummelen, Glen . 2012 . Heavenly mathematics: The forgotten art of spherical trigonometry . Princeton University Press . 98. 2012hmfa.book.....V .