Local consistency

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.

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:

$\cos(c)=\cos(a)\cos(b)+\sin(a)\sin(b)\cos(C).\,$

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 $C=\pi /2$, 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, (also called the cosine rule for angles) 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.

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 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 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.

Proof without vectors

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 $\tan ^{2}+1$),

$\tan ^{2}a+\tan ^{2}b-2\tan a\tan b\cos C\;\;\;=\;\;\;\sec ^{2}a+\sec ^{2}b-2\sec a\sec b\cos c$

$\;\;=\;\;2+\tan ^{2}a+\tan ^{2}b-2\sec a\sec b\cos c$

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

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:

$O(c^{4})+O(a^{3}b)+O(ab^{3}).\,\!$

See also

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