Napier's Rules for Right Angled Spherical Triangles
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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 a right angle.
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 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.
Tangents
The sine of the middle part equals the product of the tangents of the adjacent parts.
Cosines
The sine of the middle part equals the product of the cosine of the opposite parts.
Also known as
This suite of rules is also known as Napier's rules of circular parts.
Source of Name
This entry was named for John Napier.
Sources
- 1976: W.M. Smart: Textbook on Spherical Astronomy (6th ed.) ... (previous) ... (next): Chapter $\text I$. Spherical Trigonometry: $10$. Right-angled and quadrantal triangles.