Editing Spherical trigonometry (section)

===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′''}}.