Editing Spherical trigonometry (section)

==Preliminaries==
[[File:Spherical trigonometry Intersecting circles.svg|right|thumb|200px|Eight spherical triangles defined by the intersection of three great circles.]]

===Spherical polygons===
A '''spherical polygon''' is a ''[[polygon]]'' on the surface of the sphere. Its sides are [[Circular arc|arc]]s of [[great circle]]s—the spherical geometry equivalent of [[line segment]]s in [[Euclidean geometry|plane geometry]].

Such polygons may have any number of sides greater than 1. Two-sided spherical polygons—''[[spherical lune|lune]]s'', also called ''[[digon]]s'' 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.

From this point in the article, discussion will be restricted to spherical triangles, referred to simply as ''triangles''.

===Notation===
[[File:Spherical trigonometry basic triangle.svg|thumb|right|200px|The basic triangle on a unit sphere.]]
*Both vertices and angles at the vertices of a triangle are denoted by the same upper case letters {{mvar|A}}, {{mvar|B}}, and {{mvar|C}}.
*Sides are denoted by lower-case letters: {{mvar|a}}, {{mvar|b}}, and {{mvar|c}}. The sphere has a radius of 1, and so the side lengths and lower case angles are equivalent (see [[arc length]]). 
*The ''angle'' {{mvar|A}} (respectively, {{mvar|B}} and {{mvar|C}}) may be regarded either as the [[dihedral angle]] between the two planes that intersect the sphere at the ''[[vertex (geometry)|vertex]]'' {{mvar|A}}, or, equivalently, as the angle between the [[tangent]]s of the great circle arcs where they meet at the vertex.
*Angles are expressed in [[radian]]s. The angles of ''proper'' spherical triangles are (by convention) less than {{pi}}, so that <math display=block>
\pi < A + B + C < 3\pi
</math>(Todhunter,<ref name=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 {{pi}} radians.
*Sides are also expressed in radians. A side (regarded as a great circle arc) is measured by the angle that it subtends at the centre. On the unit sphere, this radian measure is numerically equal to the arc length. By convention, the sides of ''proper'' spherical triangles are less than {{pi}}, so that <math display=block>0 < a + b + c < 2\pi
</math>(Todhunter,<ref name=todhunter/> Art.22,32).
*The sphere's radius is taken as unity. For specific practical problems on a sphere of radius {{mvar|R}} the measured lengths of the sides must be divided by {{mvar|R}} before using the identities given below. Likewise, after a calculation on the unit sphere the sides {{mvar|a}}, {{mvar|b}}, and {{mvar|c}} must be multiplied by&nbsp;{{mvar|R}}.

===Polar triangles===
[[File:Spherical trigonometry polar triangle.svg|right|thumb|200px|The polar triangle {{math|△''A'B'C' ''}}]]
The '''polar triangle''' associated with a triangle {{math|△''ABC''}} is defined as follows. Consider the great circle that contains the side&nbsp;{{mvar|BC}}. 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 {{mvar|A}} is (conventionally) termed the pole of {{mvar|A}} and it is denoted by {{mvar|A'}}. The points {{mvar|B'}} and {{mvar|C'}} are defined similarly.

The triangle {{math|△''A'B'C' ''}} is the polar triangle corresponding to triangle&nbsp;{{math|△''ABC''}}. The angles and sides of the polar triangle are
given by (Todhunter,<ref name=todhunter/> Art.27) 
<math display=block>\begin{alignat}{3}
  A' &= \pi - a, &\qquad B' &= \pi - b , &\qquad C' &= \pi - c, \\
  a' &= \pi - A, & b' &= \pi - B , & c' &= \pi - C .
\end{alignat}</math>
Therefore, if any identity is proved for {{math|△''ABC''}} 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 {{math|3 × 3}} matrix {{mvar|M}} has the positions {{mvar|A}}, {{mvar|B}}, and {{mvar|C}} as its columns then the rows of the matrix inverse {{math|''M''{{isup|−1}}}}, if normalized to unit length, are the positions {{mvar|A′}}, {{mvar|B′}}, and {{mvar|C′}}.  In particular, when {{math|△''A′B′C′''}} is the polar triangle of {{math|△''ABC''}} then {{math|△''ABC''}} is the polar triangle of {{math|△''A′B′C′''}}.