Napier's Cosine Rule for Right Spherical Triangles
Theorem
Let $\triangle ABC$ be a right 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 the angle $\sphericalangle C$ be a right angle.
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 parts which do not neighbor the middle part be called opposite parts.
Then the sine of the middle part equals the product of the cosine of the opposite parts.
Proof
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 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: $5.102$
- 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