Spherical trigonometry is the branch of spherical geometry and trigonometry that deals with the metrical relationships between the sides and angles of spherical triangles, traditionally expressed using trigonometric functions. On the sphere, geodesics are great circles. Spherical trigonometry is of great importance for calculations in astronomy, geodesy, and navigation.
The origins of spherical trigonometry in Greek mathematics and the major developments in Islamic mathematics are discussed fully in History of trigonometry and Mathematics in medieval Islam. The subject came to fruition in Early Modern times with important developments by John Napier, Delambre and others. Since then, significant developments have been the application of vector methods, quaternion methods, and the use of numerical methods.
Preliminaries
Spherical polygons
A spherical polygon is a polygon on the surface of the sphere. Its sides are arcs of great circles—the spherical geometry equivalent of line segments in plane geometry.
Such polygons may have any number of sides greater than 1. Two-sided spherical polygons—lunes, also called digons or bi-angles—are bounded by two great-circle arcs: a familiar example is the curved outward-facing surface of a segment of an orange. Three arcs serve to define a spherical triangle, the principal subject of this article. Polygons with higher numbers of sides (4-sided spherical quadrilaterals, 5-sided spherical pentagons, etc.) are defined in similar manner. Analogously to their plane counterparts, spherical polygons with more than 3 sides can always be treated as the composition of spherical triangles.
One spherical polygon with interesting properties is the pentagramma mirificum, a 5-sided spherical star polygon with a right angle at every vertex.
Vertices, angles, and sides Both vertices and angles at the vertices are denoted by the same upper case letters, for example, , , and . The angle or spherical angle (respectively, and ) may be regarded either as the dihedral angle between the two planes that intersect the sphere at the vertex (respectively, and ), or, equivalently, as the angle between the tangents of the great circle arcs where they meet at the vertex. Side lengths on a unit-radius sphere are denoted by lower-case letters: , , and . The side lengths and lower case angles are equivalent when the latter are measured in radians (see arc length).
From this point in the article, discussion will be restricted to spherical triangles, referred to simply as triangles.
Spherical triangles
By convention, the sides of proper spherical triangles are less than radians, and (Todhunter, Art.22,32). The angles of proper spherical triangles are (by convention) less than radians, and (Todhunter, Art.22,32).In particular, the sum of the angles of a spherical triangle is strictly greater than the sum of the angles of a triangle defined on the Euclidean plane, which is always exactly radians. The sphere's radius is taken as unity. For specific practical problems on a sphere of radius the measured lengths of the sides must be divided by before using the identities given below. Likewise, after a calculation on the unit sphere the sides , , and must be multiplied by .
Polar triangles
The polar triangle associated with a triangle is defined as follows. Consider the great circle that contains the side . This great circle is defined by the intersection of a diametral plane with the surface. Draw the normal to that plane at the centre: it intersects the surface at two points and the point that is on the same side of the plane as is (conventionally) termed the pole of and it is denoted by . The points and are defined similarly.
The triangle is the polar triangle corresponding to triangle . The angles and sides of the polar triangle are given by (Todhunter, Art.27)
Therefore, if any identity is proved for then we can immediately derive a second identity by applying the first identity to the polar triangle by making the above substitutions. This is how the supplemental cosine equations are derived from the cosine equations. Similarly, the identities for a quadrantal triangle can be derived from those for a right-angled triangle. The polar triangle of a polar triangle is the original triangle.
If the matrix has the positions , , and as its columns then the rows of the matrix inverse , if normalized to unit length, are the positions , , and . In particular, when is the polar triangle of then is the polar triangle of .
Cosine rules and sine rules
Cosine rules
The cosine rule is the fundamental identity of spherical trigonometry: all other identities, including the sine rule, may be derived from the cosine rule:
These identities generalize the cosine rule of plane trigonometry, to which they are asymptotically equivalent in the limit of small interior angles. (On the unit sphere, if set and etc.; see Spherical law of cosines.)
Sine rules
The spherical law of sines is given by the formula
These identities approximate the sine rule of plane trigonometry when the sides are much smaller than the radius of the sphere.
Derivation of the cosine rule
The spherical cosine formulae were originally proved by elementary geometry and the planar cosine rule (Todhunter, Art.37). He also gives a derivation using simple coordinate geometry and the planar cosine rule (Art.60). The approach outlined here uses simpler vector methods. (These methods are also discussed at Spherical law of cosines.)
Consider three unit vectors drawn from the origin to the vertices of the triangle (on the unit sphere). The arc subtends an angle of magnitude at the centre and therefore . Introduce a Cartesian basis with along the -axis and in the -plane making an angle with the -axis. The vector projects to in the -plane and the angle between and the -axis is . Therefore, the three vectors have components:
The scalar product in terms of the components is
Equating the two expressions for the scalar product gives
This equation can be re-arranged to give explicit expressions for the angle in terms of the sides:
The other cosine rules are obtained by cyclic permutations.
Derivation of the sine rule
This derivation is given in Todhunter, (Art.40). From the identity and the explicit expression for given immediately above
Since the right hand side is invariant under a cyclic permutation of , , and the spherical sine rule follows immediately.
Alternative derivations There are many ways of deriving the fundamental cosine and sine rules and the other rules developed in the following sections. For example, Todhunter gives two proofs of the cosine rule (Articles 37 and 60) and two proofs of the sine rule (Articles 40 and 42). The page on Spherical law of cosines gives four different proofs of the cosine rule. Text books on geodesy and spherical astronomy give different proofs and the online resources of MathWorld provide yet more. There are even more exotic derivations, such as that of Banerjee who derives the formulae using the linear algebra of projection matrices and also quotes methods in differential geometry and the group theory of rotations.
The derivation of the cosine rule presented above has the merits of simplicity and directness and the derivation of the sine rule emphasises the fact that no separate proof is required other than the cosine rule. However, the above geometry may be used to give an independent proof of the sine rule. The scalar triple product, evaluates to in the basis shown. Similarly, in a basis oriented with the -axis along , the triple product , evaluates to . Therefore, the invariance of the triple product under cyclic permutations gives which is the first of the sine rules. See curved variations of the law of sines to see details of this derivation.
Differential variations When any three of the differentials , , , , , are known, the following equations, which are found by differentiating the cosine rule and using the sine rule, can be used to calculate the other three by elimination:
Identities
Supplemental cosine rules Applying the cosine rules to the polar triangle gives (Todhunter, Art.47), i.e. replacing by , by etc.,
Cotangent four-part formulae The six parts of a triangle may be written in cyclic order as (). The cotangent, or four-part, formulae relate two sides and two angles forming four consecutive parts around the triangle, for example () or ). In such a set there are inner and outer parts: for example in the set () the inner angle is , the inner side is , the outer angle is , the outer side is . The cotangent rule may be written as (Todhunter, Art.44)
and the six possible equations are (with the relevant set shown at right):
To prove the first formula start from the first cosine rule and on the right-hand side substitute for from the third cosine rule:
The result follows on dividing by . Similar techniques with the other two cosine rules give CT3 and CT5. The other three equations follow by applying rules 1, 3 and 5 to the polar triangle.
Half-angle and half-side formulae With and
Another twelve identities follow by cyclic permutation.
Todhunter (Art 45) derives the half angle formulas for the angles and sides in terms of the sides and angles, respectively. His book is available as an ebook in the public domain from Project Gutenberg. The first equation may be proved by using the law of cosines for side a in terms of sides b and c and angle A, by using the identity and by expressing the product of two sines as half the difference of the cosine of their angle difference angle minus the cosine of their angle sum (See sum-to-product identities). In detail:
The second formula uses the identity the third is a quotient and the remainder follow by applying the results to the polar triangle.
Delambre analogies The Delambre analogies (also called Gauss analogies) were published independently by Delambre, Gauss, and Mollweide in 1807–1809.
Another eight identities follow by cyclic permutation.
Proved by expanding the numerators and using the half angle formulae. (Todhunter, Art.54 and Delambre)
Napier's analogies
Another eight identities follow by cyclic permutation.
These identities follow by division of the Delambre formulae. (Todhunter, Art.52)
In particular, the first and second Napier formulas are useful in solving a spherical triangle when three of are given but not or . These two formulas give in terms of . This spherical triangle cannot be solved straightforwardly using only the cosine and sine laws.
Taking quotients of these yields the law of tangents, first stated by Persian mathematician Nasir al-Din al-Tusi (1201–1274),
Napier's rules for right spherical triangles
When one of the angles, say , of a spherical triangle is equal to /2 the various identities given above are considerably simplified. There are ten identities relating three elements chosen from the set , , , , and .
NapierTranslated by William Rae Macdonald (1889) The Construction of the Wonderful Canon of Logarithms. Edinburgh: William Blackwood and Sons. provided an elegant mnemonic aid for the ten independent equations: the mnemonic is called Napier's circle or Napier's pentagon (when the circle in the above figure, right, is replaced by a pentagon).
First, write the six parts of the triangle (three vertex angles, three arc angles for the sides) in the order they occur around any circuit of the triangle: for the triangle shown above left, going clockwise starting with gives . Next replace the parts that are not adjacent to (that is , , and ) by their complements and then delete the angle from the list. The remaining parts can then be drawn as five ordered, equal slices of a pentagram, or circle, as shown in the above figure (right). For any choice of three contiguous parts, one (the middle part) will be adjacent to two parts and opposite the other two parts. The ten Napier's Rules are given by