Napier's Tangent Rule for Quadrantal Triangles

Theorem
Let $\triangle ABC$ be a quadrantal 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 the side $c$ be a right angle.

Let the remaining parts of $\triangle ABC$ be arranged according to the exterior of this circle, where the symbol $\Box$ denotes a right angle.


 * NapiersRules.png

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.

Then the sine of the middle part equals the product of the tangents of the adjacent parts.

Proof
Let $\triangle ABC$ be a quadrantal triangle on the surface of a sphere whose center is $O$ such that side $c$ is a right angle.


 * Quadrantal-spherical-triangle.png

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

Also see

 * Napier's Cosine Rule for Quadrantal Triangles


 * Napier's Tangent Rule for Right Spherical Triangles
 * Napier's Cosine Rule for Right Spherical Triangles