Napier's Tangent Rule for Quadrantal Triangles

Theorem

 * 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 $\triangle ABC$ be quadrantal triangle such that the side $c$ is a right angle.


 * Quadrantal-spherical-triangle.png

Let the remaining parts of $\triangle ABC$ be arranged in the order as shown above:
 * $A, b, C, a, B$

Let these quantities be arranged according to the exterior of the circle above, 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.


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