Napier's Tangent Rule for Right Spherical Triangles
Theorem
Let $\triangle ABC$ be a right spherical triangle on the surface of a sphere whose center is $O$ such that the angle $\sphericalangle C$ is a right angle.
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 the remaining parts of $\triangle ABC$ be arranged according to the interior 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.
Then the sine of the middle part equals the product of the tangents of the adjacent parts.
Proof
We are given that $\sphericalangle C$ 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 Angles 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 |
$\blacksquare$
Also see
Source of Name
This entry was named for John Napier.
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 5$: Trigonometric Functions: $5.101$
- 1976: W.M. Smart: Textbook on Spherical Astronomy (6th ed.) ... (previous) ... (next): Chapter $\text I$. Spherical Trigonometry: $10$. Right-angled and quadrantal triangles.
- 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
- 2009: Murray R. Spiegel, Seymour Lipschutz and John Liu: Mathematical Handbook of Formulas and Tables (3rd ed.) ... (previous) ... (next): $\S 12$: Trigonometric Functions: $12.101.$