Napier's Rules for Quadrantal Triangles

Theorem

are the special cases of the Spherical Law of Cosines for a quadrantal triangle.

Recall the definition of Quadrantal Spherical Triangle:

Let $\triangle ABC$ be a spherical triangle.

Let one of the sides of $\triangle ABC$ be a right angle: $\dfrac \pi 2$.

Then $\triangle ABC$ is known as a quadrantal spherical triangle.

Hence, 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 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.

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.

Tangents

We are given that $c$ is a right angle.

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.

$\sin A$

$\ds \cos A \cos c$

$=$

$\ds \sin A \cot B - \sin c \cot b$

Four-Parts Formula on $b, A, c, B$

$\ds \leadsto \ \ $

$\ds \cos A \times 0$

$=$

$\ds \sin A \cot B - 1 \times \cot b$

Cosine of Right Angle, Sine of Right Angle as $c = \Box$

$\ds \leadsto \ \ $

$\ds \sin A \cot B$

$=$

$\ds \cot b$

$\ds \leadsto \ \ $

$\ds \sin A$

$=$

$\ds \tan B \cot b$

multiplying both sides by $\tan B = \dfrac 1 {\cot B}$

$\ds \leadsto \ \ $

$\ds \sin A$

$=$

$\ds \tan B \, \map \tan {\Box - b}$

Tangent of Complement equals Cotangent

$\Box$

$\sin B$

$\ds \cos B \cos c$

$=$

$\ds \sin B \cot A - \sin c \cot a$

Four-Parts Formula on $A, c, B, a$

$\ds \leadsto \ \ $

$\ds \cos B \times 0$

$=$

$\ds \sin B \cot A - 1 \times \cot a$

Cosine of Right Angle, Sine of Right Angle as $c = \Box$

$\ds \leadsto \ \ $

$\ds \sin B \cot A$

$=$

$\ds \cot a$

$\ds \leadsto \ \ $

$\ds \sin B$

$=$

$\ds \tan A \cot a$

multiplying both sides by $\tan A = \dfrac 1 {\cot A}$

$\ds \leadsto \ \ $

$\ds \sin B$

$=$

$\ds \tan A \, \map \tan {\Box - a}$

Tangent of Complement equals Cotangent

$\Box$

$\map \sin {\Box - a}$

$\ds \sin A \cos c$

$=$

$\ds \cos C \sin B + \sin C \cos B \cos a$

Analogue Formula for Spherical Law of Cosines:Corollary for side $a$

$\ds \leadsto \ \ $

$\ds \sin A \times 0$

$=$

$\ds \cos C \sin B + \sin C \cos B \cos a$

Cosine of Right Angle as $c = \Box$

$\ds \leadsto \ \ $

$\ds \sin C \cos B \cos a$

$=$

$\ds -\cos C \sin B$

$\ds \leadsto \ \ $

$\ds \cos a$

$=$

$\ds -\cot C \tan B$

dividing both sides by $\sin C \cos B$

$\ds \leadsto \ \ $

$\ds \map \sin {\Box - a}$

$=$

$\ds \map \tan {C - \Box} \tan b$

Sine of Complement equals Cosine, Sine Function is Odd, Tangent of Complement equals Cotangent

$\Box$

$\map \sin {C - \Box}$

$\ds \cos c$

$=$

$\ds \cos a \cos b + \sin a \sin b \cos C$

Spherical Law of Cosines for side $c$

$\ds \leadsto \ \ $

$\ds 0$

$=$

$\ds \cos a \cos b + \sin a \sin b \cos C$

Cosine of Right Angle as $c = \Box$

$\ds \leadsto \ \ $

$\ds \sin a \sin b \cos C$

$=$

$\ds -\cos a \cos b$

$\ds \leadsto \ \ $

$\ds -\cos C$

$=$

$\ds \cot a \cot b$

dividing both sides by $-\sin a \sin b$

$\ds \leadsto \ \ $

$\ds \map \sin {C - \Box}$

$=$

$\ds \map \tan {\Box - a} \, \map \tan {\Box - b}$

Sine of Complement equals Cosine, Sine Function is Odd, Tangent of Complement equals Cotangent

$\Box$

$\map \sin {\Box - b}$

$\ds \sin B \cos c$

$=$

$\ds \cos C \sin A + \sin C \cos A \cos b$

Analogue Formula for Spherical Law of Cosines:Corollary for side $b$

$\ds \leadsto \ \ $

$\ds \sin B \times 0$

$=$

$\ds \cos C \sin A + \sin C \cos A \cos b$

Cosine of Right Angle as $c = \Box$

$\ds \leadsto \ \ $

$\ds \sin C \cos A \cos b$

$=$

$\ds -\cos C \sin A$

$\ds \leadsto \ \ $

$\ds \cos b$

$=$

$\ds -\cot C \tan A$

dividing both sides by $\sin C \cos A$

$\ds \leadsto \ \ $

$\ds \map \sin {\Box - b}$

$=$

$\ds \map \tan {C - \Box} \tan a$

Sine of Complement equals Cosine, Sine Function is Odd, Tangent of Complement equals Cotangent

$\blacksquare$

Cosines

We are given that $c$ is a right angle.

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.

$\sin A$

$\ds \dfrac {\sin A} {\sin a}$

$=$

$\ds \dfrac {\sin C} {\sin c}$

Spherical Law of Sines for angle $A$

$\ds \leadsto \ \ $

$\ds \dfrac {\sin A} {\sin a}$

$=$

$\ds \dfrac {\sin C} 1$

Sine of Right Angle as $c = \Box$

$\ds \leadsto \ \ $

$\ds \sin A$

$=$

$\ds \sin a \sin C$

$\ds \leadsto \ \ $

$\ds \sin A$

$=$

$\ds \map \cos {\Box - a} \, \map \cos {\Box - C}$

Cosine of Complement equals Sine

$\ds $

$=$

$\ds \map \cos {\Box - a} \, \map \cos {C - \Box}$

Cosine Function is Even

$\Box$

$\sin B$

$\ds \dfrac {\sin B} {\sin b}$

$=$

$\ds \dfrac {\sin C} {\sin c}$

Spherical Law of Sines for angle $B$

$\ds \leadsto \ \ $

$\ds \dfrac {\sin B} {\sin b}$

$=$

$\ds \dfrac {\sin C} 1$

Sine of Right Angle as $c = \Box$

$\ds \leadsto \ \ $

$\ds \sin B$

$=$

$\ds \sin b \sin C$

$\ds \leadsto \ \ $

$\ds \sin B$

$=$

$\ds \map \cos {\Box - b} \, \map \cos {\Box - C}$

Cosine of Complement equals Sine

$\ds $

$=$

$\ds \map \cos {\Box - b} \, \map \cos {C - \Box}$

Cosine Function is Even

$\Box$

$\map \sin {\Box - a}$

$\ds \cos a$

$=$

$\ds \cos b \cos c + \sin b \sin c \cos A$

Spherical Law of Cosines for side $a$

$\ds $

$=$

$\ds \cos b \times 0 + \sin b \times 1 \times \cos A$

Cosine of Right Angle and Sine of Right Angle as $c = \Box$

$\ds $

$=$

$\ds \sin b \cos A$

$\ds \leadsto \ \ $

$\ds \map \sin {\Box - a}$

$=$

$\ds \map \cos {\Box - b} \cos A$

Sine of Complement equals Cosine, Cosine of Complement equals Sine

$\Box$

$\map \sin {C - \Box}$

$\ds \cos C$

$=$

$\ds -\cos A \cos B + \sin A \sin B \cos c$

Spherical Law of Cosines for angle $C$

$\ds \leadsto \ \ $

$\ds -\cos C$

$=$

$\ds \cos A \cos B$

Cosine of Right Angle as $c = \Box$

$\ds \leadsto \ \ $

$\ds \map \sin {C - \Box}$

$=$

$\ds \cos A \cos B$

Sine of Complement equals Cosine and Sine Function is Odd

$\Box$

$\map \sin {\Box - b}$

$\ds \cos b$

$=$

$\ds \cos a \cos c + \sin a \sin c \cos B$

Spherical Law of Cosines for side $b$

$\ds $

$=$

$\ds \cos a \times 0 + \sin a \times 1 \cos B$

Cosine of Right Angle as $c = \Box$

$\ds $

$=$

$\ds \sin a \cos B$

$\ds \leadsto \ \ $

$\ds \map \sin {\Box - b}$

$=$

$\ds \map \cos {\Box - a} \cos B$

Sine of Complement equals Cosine, Cosine of Complement equals Sine

$\blacksquare$

Also see

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.