Napier's Rules for Quadrantal Triangles

Theorem
Napier's Rules for Quadrantal Triangles are the special cases of the Spherical Law of Cosines for a spherical triangle one of whose sides is a 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 side $c$ be a right angle.

Let the remaining parts of $\triangle ABC$ be arranged in a circle as the exterior of the above, where the symbol $\Box$ denotes a right angle.

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 sine of the middle part equals the product of the tangents of the adjacent parts.
 * The sine of the middle part equals the product of the cosines of the opposite parts.