Spherical law of cosines

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:Romuald Ireneus 'Scibor-Marchocki
Spherical trigonometry
''Elementary-Geometry Trigonometry'' web page (1997).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).

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

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:

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

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.

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

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

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 $\mathbf$ is at the north pole and $\mathbf$ is somewhere on the prime meridian (longitude of 0). With this rotation, the spherical coordinates for $\mathbf$ 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 $\mathbf$ are $(r, \theta, \phi) = (1, b, C) .$ The Cartesian coordinates for $\mathbf$ are $(x, y, z) = (\sin a, 0, \cos a)$ and the Cartesian coordinates for $\mathbf$ 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
$\begin \sin a \sin b \cos C &= (\bold u \times \bold v) \cdot (\bold u \times \bold w) \\ &= (\bold u \cdot \bold u)(\bold v \cdot \bold w) - (\bold u \cdot \bold w)(\bold v \cdot \bold u) \\ &= \cos c - \cos a \cos b \end$

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

Third proof

The following proof relies on the concept of quaternions and is based on a proof given in Brand: Let , , and denote the unit vectors from the center of the unit sphere to those corners of the triangle. We define the quaternion . The quaternion is used to represent a rotation by 180° around the axis indicated by the vector . We note that using as the axis of rotation gives the same result, and that the rotation is its own inverse. We also define and .

We compute the product of quaternions, which also gives the composition of the corresponding rotations:
:
where represents the real (scalar) and imaginary (vector) parts of a quaternion, is the angle between and , and is the axis of the rotation that moves to along a great circle. Similarly we define:
:.
:
The quaternions , , and are used to represent rotations with axes of rotation , , and , respectively, and angles of rotation , , and , respectively. (Because these are double angles, each of , , and represents two applications of the rotation implied by an edge of the spherical triangle.)

From the definitions, it follows that
:,
which tells us that the composition of these rotations is the identity transformation. In particular, gives us
:.

Expanding the left-hand side, we obtain

:$(\cos a \cos b - \bold u' \cdot \bold w' \sin a \sin b, \bold u' \cos a \sin b + \bold w' \sin a \cos b + \bold u' \times \bold w' \sin a \sin b).$

Equating the real parts on both sides of the identity, we obtain

:$\cos a \cos b - \bold u' \cdot \bold w' \sin a \sin b = \cos c.$

Because is parallel to , is parallel to , and is the angle between and , it follows that $\bold u' \cdot \bold w' = -\cos C$. 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 () and angles () on opposite sides of the equations:
$\begin \cos C &= \frac \\ \cos c &= \frac \\ \end$

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,
$c^2 \approx a^2 + b^2 - 2ab\cos C \,.$

To prove this, we will use the small-angle approximation obtained from the Maclaurin series for the cosine and sine functions:
$\begin \cos a &= 1 - \frac + O\left(a^4\right) \\ \sin a &= a + O\left(a^3\right) \end$

Substituting these expressions into the spherical law of cosines nets:

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

or after simplifying:

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

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

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

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 of the Sun in terms of its altitude , terrestrial latitude , and ortive amplitude (angular distance between due East and the Sun's rising place on the horizon) as . (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 ''De triangulis omnimodis'' (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, , rather than the cosine,

:$\frac = \frac{\sin a \sin b}.$

Mathematical historians have speculated that Regiomontanus may have adapted the result from specific astronomical examples in al-Battānī's ''Kitāb az-Zīj aṣ-Ṣābi’'', which was published in Latin translation annotated by Regiomontanus in 1537.

See also

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

Notes

Spherical trigonometry
Articles containing proofs
Theorems in geometry

he:טריגונומטריה ספירית#משפט הקוסינוסים