Napier's Rules for Right Angled Spherical Triangles

Theorem
Napier's Rules for Right Angled Spherical Triangles are the special cases of the Spherical Law of Cosines for a spherical triangle one of whose angles or sides is right angle.


 * NapiersRules.png

Let $\triangle ABC$ be a spherical triangle on the surface of a sphere whose center is $O$.

Let the sides $a, b, c$ of $\triangle ABC$ be measured by the angles subtended at $O$, where $a, b, c$ are opposite $A, B, C$ respectively.

Let either angle $\angle C$ or side $c$ be a right angle.

Let the remaining parts of $\triangle ABC$ be arranged in a circle as above:
 * for $\angle C$ a right angle, the interior
 * for $c$ a right angle, the exterior

where the prefix $\text{co-}$ meaning complement is added to $c, A, B$.

Let one of the parts of this circle be called a middle part.

Let the two neighboring parts of the middle part be called adjacent parts.

Let the remaining two parts be called opposite parts.

Then the following relations are valid: