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.

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.

Tangents

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.

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

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..

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

$\blacksquare$

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.