Editing Spherical trigonometry (section)

===Oblique triangles===
The solution of triangles is the principal purpose of spherical trigonometry: given three, four or five elements of the triangle, determine the others. The case of five given elements is trivial, requiring only a single application of the sine rule. For four given elements there is one non-trivial case, which is discussed below. For three given elements there are six cases: three sides, two sides and an included or opposite angle, two angles and an included or opposite side, or three angles. (The last case has no analogue in planar trigonometry.) No single method solves all cases. The figure below shows the seven non-trivial cases: in each case the given sides are marked with a cross-bar and the given angles with an arc. (The given elements are also listed below the triangle).  In the summary notation here such as ASA, A refers to a given angle and S refers to a given side, and the sequence of A's and S's in the notation refers to the corresponding sequence in the triangle.
[[File:Spherical trigonometry triangle cases.svg|thumb|center|500px]]

*'''Case 1: three sides given (SSS).'''  The cosine rule may be used to give the angles {{mvar|A}}, {{mvar|B}}, and {{mvar|C}} but, to avoid ambiguities, the half angle formulae are preferred.
*'''Case 2: two sides and an included angle given (SAS).''' The cosine rule gives {{mvar|a}} and then we are back to Case&nbsp;1.
*'''Case 3: two sides and an opposite angle given (SSA).''' The sine rule gives {{mvar|C}} and then we have Case&nbsp;7. There are either one or two solutions.
*'''Case 4: two angles and an included side given (ASA).''' The four-part cotangent formulae for sets ({{mvar|cBaC}}) and ({{mvar|BaCb}}) give {{mvar|c}} and {{mvar|b}}, then {{mvar|A}} follows from the sine rule.
*'''Case 5: two angles and an opposite side given (AAS).''' The sine rule gives {{mvar|b}} and then we have Case&nbsp;7 (rotated). There are either one or two solutions.
*'''Case 6: three angles given (AAA).''' The supplemental cosine rule may be used to give the sides {{mvar|a}}, {{mvar|b}}, and {{mvar|c}} but, to avoid ambiguities, the half-side formulae are preferred.
*'''Case 7: two angles and two opposite sides given (SSAA).''' Use Napier's analogies for {{mvar|a}} and {{mvar|A}}; or, use Case 3 (SSA) or case 5 (AAS).

The solution methods listed here are not the only possible choices: many others are possible. In general it is better to choose methods that avoid taking an inverse sine because of the possible ambiguity between an angle and its supplement. The use of half-angle formulae is often advisable because half-angles will be less than {{pi}}/2 and therefore free from ambiguity. There is a full discussion in Todhunter. The article [[Solution of triangles#Solving spherical triangles]] presents variants on these methods with a slightly different notation.

There is a full discussion of the solution of oblique triangles in Todhunter.<ref name=todhunter/>{{rp|Chap. VI}} See also the discussion in Ross.<ref>Ross, Debra Anne. ''Master Math: Trigonometry'', Career Press, 2002.</ref> [[Nasir al-Din al-Tusi]] was the first to list the six distinct cases (2–7 in the diagram) of a right triangle in spherical trigonometry.<ref>{{MacTutor|id=Al-Tusi_Nasir|title=Nasir al-Din al-Tusi}} "One of al-Tusi's most important mathematical contributions was the creation of trigonometry as a mathematical discipline in its own right rather than as just a tool for astronomical applications. In Treatise on the quadrilateral al-Tusi gave the first extant exposition of the whole system of plane and spherical trigonometry. This work is really the first in history on trigonometry as an independent branch of pure mathematics and the first in which all six cases for a right-angled spherical triangle are set forth"</ref>

[[File:Spherical trigonometry solution construction.svg|thumb|100px]]