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.