Spherical Law Of Cosines
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In
spherical trigonometry Spherical trigonometry is the branch of spherical geometry that deals with the metrical relationships between the sides and angles of spherical triangles, traditionally expressed using trigonometric functions. On the sphere, geodesics are gr ...
, the law of cosines (also called the cosine rule for sides) is a theorem relating the sides and angles of
spherical triangles Spherical trigonometry is the branch of spherical geometry that deals with the metrical relationships between the sides and angles of spherical triangles, traditionally expressed using trigonometric functions. On the sphere, geodesics are gr ...
, analogous to the ordinary
law of cosines In trigonometry, the law of cosines (also known as the cosine formula, cosine rule, or al-Kashi's theorem) relates the lengths of the sides of a triangle to the cosine of one of its angles. Using notation as in Fig. 1, the law of cosines states ...
from plane
trigonometry Trigonometry () is a branch of mathematics that studies relationships between side lengths and angles of triangles. The field emerged in the Hellenistic world during the 3rd century BC from applications of geometry to astronomical studies. T ...
. Given a unit sphere, a "spherical triangle" on the surface of the sphere is defined by the
great circle In mathematics, a great circle or orthodrome is the circular intersection of a sphere and a plane passing through the sphere's center point. Any arc of a great circle is a geodesic of the sphere, so that great circles in spherical geomet ...
s 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
radian The radian, denoted by the symbol rad, is the unit of angle in the International System of Units (SI) and is the standard unit of angular measure used in many areas of mathematics. The unit was formerly an SI supplementary unit (before that c ...
s) 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 In mathematics, the Pythagorean theorem or Pythagoras' theorem is a fundamental relation in Euclidean geometry between the three sides of a right triangle. It states that the area of the square whose side is the hypotenuse (the side opposite t ...
: :\cos c = \cos a \cos b\, If the law of cosines is used to solve for , the necessity of inverting the cosine magnifies
rounding error A roundoff error, also called rounding error, is the difference between the result produced by a given algorithm using exact arithmetic and the result produced by the same algorithm using finite-precision, rounded arithmetic. Rounding errors are d ...
s when is small. In this case, the alternative formulation of the
law of haversines The haversine formula determines the great-circle distance between two points on a sphere given their longitudes and latitudes. Important in navigation, it is a special case of a more general formula in spherical trigonometry, 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 vector In mathematics, a unit vector in a normed vector space is a vector (often a spatial vector) of length 1. A unit vector is often denoted by a lowercase letter with a circumflex, or "hat", as in \hat (pronounced "v-hat"). The term ''direction vecto ...
s 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 The North Pole, also known as the Geographic North Pole or Terrestrial North Pole, is the point in the Northern Hemisphere where the Earth's axis of rotation meets its surface. It is called the True North Pole to distinguish from the Mag ...
and \mathbf is somewhere on the
prime meridian A prime meridian is an arbitrary meridian (a line of longitude) in a geographic coordinate system at which longitude is defined to be 0°. Together, a prime meridian and its anti-meridian (the 180th meridian in a 360°-system) form a great c ...
(longitude of 0). With this rotation, the
spherical coordinates In mathematics, a spherical coordinate system is a coordinate system for three-dimensional space where the position of a point is specified by three numbers: the ''radial distance'' of that point from a fixed origin, its ''polar angle'' measu ...
for \mathbf are (r, \theta, \phi) = (1, a, 0), where ''θ'' 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 vector In mathematics, a unit vector in a normed vector space is a vector (often a spatial vector) of length 1. A unit vector is often denoted by a lowercase letter with a circumflex, or "hat", as in \hat (pronounced "v-hat"). The term ''direction vecto ...
s 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 product In mathematics, the cross product or vector product (occasionally directed area product, to emphasize its geometric significance) is a binary operation on two vectors in a three-dimensional oriented Euclidean vector space (named here E), and is ...
s,
dot product In mathematics, the dot product or scalar productThe term ''scalar product'' means literally "product with a scalar as a result". It is also used sometimes for other symmetric bilinear forms, for example in a pseudo-Euclidean space. is an algebra ...
s, and the
Binet–Cauchy identity In algebra, the Binet–Cauchy identity, named after Jacques Philippe Marie Binet and Augustin-Louis Cauchy, states that \left(\sum_^n a_i c_i\right) \left(\sum_^n b_j d_j\right) = \left(\sum_^n a_i d_i\right) \left(\sum_^n b_j c_j\right) + \su ...
.


Third proof

Let , and denote the
unit vector In mathematics, a unit vector in a normed vector space is a vector (often a spatial vector) of length 1. A unit vector is often denoted by a lowercase letter with a circumflex, or "hat", as in \hat (pronounced "v-hat"). The term ''direction vecto ...
s 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 another rotation of vector to by an angle b, after which we rotate the vector back to by an angle c. The composition of these three rotations will form an identity transform. That is, the composite rotation maps the point to itself. These three rotational operations can be represented by
quaternions In mathematics, the quaternion number system extends the complex numbers. Quaternions were first described by the Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. Hamilton defined a quater ...
: : \begin q_A &= \cos \frac + \mathbf \sin \frac, \\ q_B &= \cos \frac + \mathbf \sin \frac, \\ q_C &= \cos \frac + \mathbf \sin \frac, \end where \mathbf, \mathbf, and \mathbf are the unit vectors representing the axes of rotations, as defined by the
right-hand rule In mathematics and physics, the right-hand rule is a common mnemonic for understanding orientation of axes in three-dimensional space. It is also a convenient method for quickly finding the direction of a cross-product of 2 vectors. Most of th ...
, respectively. The composition of these three rotations 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^* = \cos \frac - \mathbf \sin \frac and q_B^* = \cos \frac - \mathbf \sin \frac. This gives us the identity :\cos \frac + \mathbf \sin \frac = \left(\cos \frac - \mathbf \sin \frac\right) \left( \cos \frac - \mathbf \sin \frac \right). The quaternion product on the right-hand side of this identity is given by :\left(\cos \frac \cos \frac - \mathbf \cdot \mathbf \sin \frac \sin \frac \right) - \left(\mathbf \sin \frac \cos \frac + \mathbf \cos \frac \sin \frac - \mathbf \times \mathbf \sin \frac \sin \frac \right). Equating the scalar parts on both sides of the identity, we have :\cos \frac = \cos \frac \cos \frac - \mathbf \cdot \mathbf \sin \frac \sin \frac. Here \mathbf \cdot \mathbf = \cos (\pi - C) = - \cos C. Since this identity is valid for any angles, suppressing the halves, we have :\cos c = \cos a \cos b + \cos C \sin a \sin b. We can also recover the sine law by first noting that \mathbf \times \mathbf = -\mathbf \sin C and then equating the vector parts on both sides of the identity as :\mathbf \sin \frac = -\left( \mathbf \sin \frac \cos \frac + \mathbf \cos \frac \sin \frac + \mathbf \sin C \sin \frac \sin \frac \right). The vector \mathbf is orthogonal to both the vectors \mathbf and \mathbf, and as such \mathbf \cdot \mathbf = \mathbf \cdot \mathbf = 0. Taking dot product with respect to \mathbf on both sides, and suppressing the halves, we have \mathbf \cdot \mathbf \sin c = -\sin C \sin a \sin b. Now \mathbf \times \mathbf = -\mathbf \sin c and so we have \mathbf \cdot (\mathbf \times \mathbf) = -\mathbf \cdot \mathbf \sin c = \sin C \sin a \sin b. Dividing each side by \sin a \sin b \sin c, we have :\frac = \frac. Since the right-hand side of the above expression is unchanged by cyclic permutation, we have :\frac = \frac = \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: :\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 The small-angle approximations can be used to approximate the values of the main trigonometric functions, provided that the angle in question is small and is measured in radians: : \begin \sin \theta &\approx \theta \\ \cos \theta &\approx 1 - \ ...
obtained from the
Maclaurin series Maclaurin or MacLaurin is a surname. Notable people with the surname include: * Colin Maclaurin (1698–1746), Scottish mathematician * Normand MacLaurin (1835–1914), Australian politician and university administrator * Henry Normand MacLaurin ( ...
for the cosine and sine functions: : \cos a = 1 - \frac + O\left(a^4\right),\, \sin a = a + O\left(a^3\right) 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 Big O or The Big O may refer to: Fiction * ''The Big O'', a 1999 Japanese animated TV series Mathematics and computing * Big Omega function (disambiguation), various arithmetic functions in number theory * Big O notation, asymptotic behavior in ...
terms for and are dominated by as and {{math, ''b'' 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).


See also

*
Half-side formula In spherical trigonometry, the half side formula relates the angles and lengths of the sides of spherical triangles, which are triangles drawn on the surface of a sphere and so have curved sides and do not obey the formulas for plane triangles. Fo ...
*
Hyperbolic law of cosines In hyperbolic geometry, the "law of cosines" is a pair of theorems relating the sides and angles of triangles on a hyperbolic plane, analogous to the planar law of cosines from plane trigonometry, or the spherical law of cosines in spherical trigono ...
*
Solution of triangles Solution of triangles ( la, solutio triangulorum) is the main trigonometric problem of finding the characteristics of a triangle (angles and lengths of sides), when some of these are known. The triangle can be located on a plane or on a sphere. Appl ...
*
Spherical law of sines In trigonometry, the law of sines, sine law, sine formula, or sine rule is an equation relating the lengths of the sides of any triangle to the sines of its angles. According to the law, \frac \,=\, \frac \,=\, \frac \,=\, 2R, where , and ...


Notes

Spherical trigonometry Articles containing proofs Theorems in geometry he:טריגונומטריה ספירית#משפט הקוסינוסים