Napier's Rules for Right Spherical Triangles
Theorem
Napier's Rules for Right Spherical Triangles are the special cases of the Spherical Law of Cosines for a spherical triangle one of whose angles 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 angle $\angle C$ be a right angle.
Let the remaining parts of $\triangle ABC$ be arranged in a circle as the interior 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 right spherical triangle such that the angle $\sphericalangle C$ is a right angle.
Let the remaining parts of $\triangle ABC$ be arranged according to the interior 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 for angle $A$ | |||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \sin a \times 0\) | \(=\) | \(\ds \cos c \sin b - \sin c \cos b \cos A\) | 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 $\sin c \cos b$ | ||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \map \sin {\Box - A}\) | \(=\) | \(\ds \map \tan {\Box - c} \tan b\) | Sine of Complement equals Cosine, Tangent of Complement equals Cotangent |
$\Box$
$\map \sin {\Box - c}$
| \(\ds \cos C\) | \(=\) | \(\ds -\cos A \cos B + \sin A \sin B \cos c\) | Spherical Law of Cosines for angle $C$ | |||||||||||
| \(\ds \leadsto \ \ \) | \(\ds 0\) | \(=\) | \(\ds -\cos A \cos B + \sin A \sin B \cos c\) | 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 $\sin A \sin B$ | ||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \map \sin {\Box - c}\) | \(=\) | \(\ds \map \tan {\Box - A} \, \map \tan {\Box - B}\) | Sine of Complement equals Cosine, 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 for angle $B$ | |||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \sin b \times 0\) | \(=\) | \(\ds \cos c \sin a - \sin c \cos a \cos B\) | 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 $\sin c \cos a$ | ||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \map \sin {\Box - B}\) | \(=\) | \(\ds \map \tan {\Box - c} \tan a\) | Sine of Complement equals Cosine, Tangent of Complement equals Cotangent |
$\blacksquare$
Cosines
Let $\triangle ABC$ be a right spherical triangle such that the angle $\sphericalangle C$ is a right angle.
Let the remaining parts of $\triangle ABC$ be arranged according to the interior 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 side $a$ | |||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \dfrac {\sin a} {\sin A}\) | \(=\) | \(\ds \dfrac {\sin c} 1\) | 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 |
$\Box$
$\sin b$
| \(\ds \dfrac {\sin b} {\sin B}\) | \(=\) | \(\ds \dfrac {\sin c} {\sin C}\) | Spherical Law of Sines for side $b$ | |||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \dfrac {\sin b} {\sin B}\) | \(=\) | \(\ds \dfrac {\sin c} 1\) | 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 |
$\Box$
$\map \sin {\Box - A}$
| \(\ds \cos A\) | \(=\) | \(\ds -\cos B \cos C + \sin B \sin C \cos a\) | Spherical Law of Cosines for angle $A$ | |||||||||||
| \(\ds \) | \(=\) | \(\ds -\cos B \times 0 + \sin B \times 1 \times \cos a\) | 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 |
$\Box$
$\map \sin {\Box - c}$
| \(\ds \cos c\) | \(=\) | \(\ds \cos a \cos b + \sin a \sin b \cos C\) | Spherical Law of Cosines for side $c$ | |||||||||||
| \(\ds \) | \(=\) | \(\ds \cos a \cos b\) | Cosine of Right Angle as $C = \Box$ | |||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \map \sin {\Box - c}\) | \(=\) | \(\ds \cos a \cos b\) | Sine of Complement equals Cosine |
$\Box$
$\map \sin {\Box - B}$
| \(\ds \cos B\) | \(=\) | \(\ds -\cos A \cos C + \sin A \sin C \cos b\) | Spherical Law of Cosines for angle $B$ | |||||||||||
| \(\ds \) | \(=\) | \(\ds -\cos A \times 0 + \sin A \times 1 \cos b\) | Cosine of Right Angle as $C = \Box$ | |||||||||||
| \(\ds \) | \(=\) | \(\ds \sin A \cos b\) | ||||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \map \sin {\Box - B}\) | \(=\) | \(\ds \map \cos {\Box - A} \cos c\) | Sine of Complement equals Cosine, Cosine of Complement equals Sine |
$\blacksquare$
Also known as
These rules are 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.
- 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): Napier, John (1550-1617)
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): Napier's rules of circular parts
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): Napier's rules of circular parts