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.

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, the lengths are the subtended angles times the radius, and the formula still holds if $a, b$ and $c$ are reinterpreted as the subtended angles). As a special case, for $C = π / 2$ , then $cos C = 0$ , and one obtains the spherical analogue of the Pythagorean theorem:

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

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.^[3]

A variation on the law of cosines, the second spherical law of cosines,^[4] (also called the cosine rule for angles^[1]) states:

\cos C=-\cos A\cos B+\sin A\sin B\cos c\,,

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.

Proofs

First 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 $t a$ 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 {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 $sin 2 a = 1 - cos 2 a$ 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.

Second proof

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 $sec 2$ of any angle is $tan 2 + 1$ ),

{\begin{aligned}\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{aligned}}

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

Multiply both sides by $cos a cos b$ and rearrange.

Third proof

The angles and distances do not change if the sphere is rotated, so we can rotate the sphere so that $\mathbf {u}$ is at the north pole and $\mathbf {v}$ is somewhere on the prime meridian (longitude of 0). With this rotation, the spherical coordinates for $\mathbf {v}$ are $(r,\theta ,\phi )=(1,a,0)$ and the spherical coordinates for $\mathbf {w}$ are $(r,\theta ,\phi )=(1,b,C)$ . The Cartesian coordinates for $\mathbf {v}$ are $(x,y,z)=(\sin a,0,\cos a)$ and the Cartesian coordinates for $\mathbf {w}$ 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 $\sin a\sin b\cos C+\cos a\cos b$ .

Rearrangements

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

{\begin{aligned}\cos C&={\frac {\cos c-\cos a\cos b}{\sin a\sin b}}\\\cos c&={\frac {\cos C+\cos A\cos B}{\sin A\sin B}}\end{aligned}}

Planar limit: small angles

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 will 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(ab^{3}).\,\!

Notes

1 2 3 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).
1 2 Romuald Ireneus 'Scibor-Marchocki, Spherical trigonometry, Elementary-Geometry Trigonometry web page (1997).
↑ R. W. Sinnott, "Virtues of the Haversine", Sky and Telescope 68 (2), 159 (1984).
↑ Reiman, István (1999). Geometria és határterületei. Szalay Könyvkiadó és Kereskedőház Kft. p. 83.

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