Spherical law of cosines

Template:Short description In spherical trigonometry, the law of cosines (or, more specifically, the law of cosines for sides) is a theorem relating the three sides and one of the angles of a spherical triangle, analogous to the planar law of cosines for a triangle in the Euclidean plane.

Spherical triangle solved by the law of cosines.

A spherical triangle is a shape on a sphere consisting of three vertices (corner points) connected by three sides, each of which is part of a great circle, the analog on the sphere of a straight line in the plane (for example the equator and meridians of a globe). The arc lengths of the sides are proportional to the measures of the central angles they subtend. The angles between sides are dihedral angles between the planes containing them. In the image, Template:Math, Template:Math, and Template:Math represent three points on the sphere. If the angular lengths of the sides are Template:Math (from Template:Math to Template:Math (from Template:Math to Template:Math), and Template:Math (from Template:Math to Template:Math), and the angle of the corner opposite Template:Math is Template:Math, then the (first) spherical law of cosines states: $$\cos c=\cos a\cos b+\sin a\sin b\cos C$$

As a special case, when Template:Math then Template:Math, resulting in the spherical analogue of the Pythagorean theorem: $$\cos c=\cos a\cos b$$

If the law of cosines is used to solve for Template:Math, the necessity of inverting the cosine magnifies rounding errors when Template:Math is small. In this case, the alternative formulation of the law of haversines is preferable.

A second spherical law of cosines, sometimes called the law of cosines for angles, relates the three angles and one of the sides of a triangle: $$\cos C=-\cos A\cos B+\sin A\sin B\cos c$$ where Template:Math and Template:Math are the angles of the corners opposite to sides Template:Math and Template:Math, respectively. It is related to the first law by polar duality.

Proofs

First proof

Let Template:Math, and Template:Math denote the unit vectors from the center of a unit 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 $(r,\theta ,\varphi )=(1,a,0),$ where Template:Mvar is the angle measured from the north pole not from the equator, and the spherical coordinates for $𝐰$ are $(r,\theta ,\varphi )=(1,b,C).$ 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 Template:Math, and Template:Math denote the unit vectors from the center of the sphere to those corners of the triangle. We have Template:Math, Template:Math, Template:Math, and Template:Math. The vectors Template:Math and Template:Math have lengths Template:Math and Template:Math respectively and the angle between them is Template:Math, so $$\begin{array}{cc}\sin a\sin b\cos C& =(𝐮\times 𝐯)\cdot (𝐮\times 𝐰)\\ & =(𝐮\cdot 𝐮)(𝐯\cdot 𝐰)-(𝐮\cdot 𝐰)(𝐯\cdot 𝐮)\\ & =\cos c-\cos a\cos b\end{array}$$ using cross products, dot products, and the Binet–Cauchy identity $$(𝐩\times 𝐪)\cdot (𝐫\times 𝐬)=(𝐩\cdot 𝐫)(𝐪\cdot 𝐬)-(𝐩\cdot 𝐬)(𝐪\cdot 𝐫).$$

Third proof

There is a proof using quaternions. Let Template:Math, Template:Math, and Template:Math denote the unit vectors from the center of the unit sphere to those corners of the triangle. We define the quaternion Template:Math. The quaternion Template:Mvar is used to represent a rotation by 180° around the axis indicated by the vector Template:Math. We note that using Template:Math as the axis of rotation gives the same result, and that the rotation is its own inverse. We also define Template:Math and Template:Math.

We compute the product of quaternions, which also gives the composition of the corresponding rotations: $$q=v{u}^{-1}=(v)(-u)=(-(𝐯\cdot -𝐮),𝐯\times -𝐮)=(𝐮\cdot 𝐯,𝐮\times 𝐯)=(\cos a,{𝐰}^{\prime }\sin a)$$ where Template:Math represents the real (scalar) and imaginary (vector) parts of a quaternion, Template:Mvar is the angle between Template:Math and Template:Math, and Template:Math is the axis of the rotation that moves Template:Math to Template:Math along a great circle. Similarly we define: $$\begin{array}{cc}r& =w{v}^{-1}=(𝐯\cdot 𝐰,𝐯\times 𝐰)=(\cos c,{𝐮}^{\prime }\sin c)\\ s& =u{w}^{-1}=(𝐰\cdot 𝐮,𝐰\times 𝐮)=(\cos b,{𝐯}^{\prime }\sin b).\end{array}$$

The quaternions Template:Mvar, Template:Mvar, and Template:Mvar are used to represent rotations with axes of rotation Template:Math, Template:Math, and Template:Math, respectively, and angles of rotation Template:Math, Template:Math, and Template:Math, respectively. (Because these are double angles, each of Template:Mvar, Template:Mvar, and Template:Mvar represents two applications of the rotation implied by an edge of the spherical triangle.)

From the definitions, it follows that $$qsr=v{u}^{-1}u{w}^{-1}w{v}^{-1}=1,$$ which tells us that the composition of these rotations is the identity transformation. In particular, Template:Math gives us $$(\cos a,{𝐰}^{\prime }\sin a)(\cos b,{𝐯}^{\prime }\sin b)=(\cos c,-{𝐮}^{\prime }\sin c).$$

Expanding the left-hand side, we obtain $$(\cos a\cos b-{𝐰}^{\prime }\cdot {𝐯}^{\prime }\sin a\sin b,{𝐰}^{\prime }\cos a\sin b+{𝐯}^{\prime }\sin a\cos b+{𝐰}^{\prime }\times {𝐯}^{\prime }\sin a\sin b).$$

Equating the real parts on both sides of the identity, we obtain $$\cos a\cos b-{𝐰}^{\prime }\cdot {𝐯}^{\prime }\sin a\sin b=\cos c.$$

Because Template:Math is parallel to Template:Math, Template:Math is parallel to Template:Math, and Template:Mvar is the angle between Template:Math and Template:Math, it follows that ${𝐰}^{\prime }\cdot {𝐯}^{\prime }=-\cos C$. Thus, $$\cos a\cos b+\cos C\sin a\sin b=\cos c.$$

Rearrangements

The first and second spherical laws of cosines can be rearranged to put the sides (Template:Math) and angles (Template:Math) on opposite sides of the equations: $$\begin{array}{cc}\cos C& =\frac{\cos c-\cos a\cos b}{\sin a\sin b}\\ \cos c& =\frac{\cos C+\cos A\cos B}{\sin A\sin B}\end{array}$$

Planar limit: small angles

For small spherical triangles, i.e. for small Template:Math, and Template:Math, 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: $$\begin{array}{cc}\cos a& =1-\frac{a^2}{2}+O\left(a^4\right)\\ \sin a& =a+O\left(a^3\right)\end{array}$$

Substituting these expressions into the spherical law of cosines nets: $$1-\frac{c^2}{2}+O\left(c^4\right)=1-\frac{a^2}{2}-\frac{b^2}{2}+\frac{a^2{b}^2}{4}+O\left(a^4\right)+O\left(b^4\right)+\cos (C)(ab+O\left(a^{3}b\right)+O\left(a{b}^3\right)+O\left(a^3{b}^3\right))$$ or after simplifying: $$c^2=a^2+b^2-2ab\cos C+O\left(c^4\right)+O\left(a^4\right)+O\left(b^4\right)+O\left(a^2{b}^2\right)+O\left(a^{3}b\right)+O\left(a{b}^3\right)+O\left(a^3{b}^3\right).$$

The big O terms for Template:Math and Template:Math are dominated by Template:Math as Template:Math and Template:Math get small, so we can write this last expression as: $$c^2=a^2+b^2-2ab\cos C+O\left(a^4\right)+O\left(b^4\right)+O\left(c^4\right).$$

History

Various trigonometric equations equivalent to the spherical law of cosines were used in the course of solving astronomical problems by medieval Islamic astronomers al-Khwārizmī (9th century) and al-Battānī (c. 900), Indian astronomer Nīlakaṇṭha (15th century), and Austrian astronomer Georg von Peuerbach (15th century) but none of them treated it as a general method for solving spherical triangles. For example, al-Khwārizmī calculated the azimuth Template:Tmath of the Sun in terms of its altitude Template:Tmath, terrestrial latitude Template:Tmath, and ortive amplitude Template:Tmath (angular distance between due East and the Sun's rising place on the horizon) as Template:Tmath.Template:Sfn (See Horizontal coordinate system.)

The spherical law of cosines appeared as an independent trigonometrical identity for solving spherical triangles in Peuerbach's student Regiomontanus's Template:Lang (unfinished at Regiomontanus's death in 1476, published posthumously 1533), a foundational work for European trigonometry and astronomy which comprehensively described how to solve plane and spherical triangles. Regiomontanus used nearly the modern form, but written in terms of the versine, Template:Tmath, rather than the cosine,Template:Sfn $$\frac{\mathrm{vers}C}{\mathrm{vers}c-\mathrm{vers}(a-b)}=\frac{1}{\sin a\sin b}.$$

Mathematical historians have speculated that Regiomontanus may have adapted the result from specific astronomical examples in al-Battānī's Template:Transliteration, which was published in Latin translation annotated by Regiomontanus in 1537.Template:Sfn

See also

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

References

• Template:Cite book