Napier's Cosine Rule for Quadrantal Triangles
Theorem
Let $\triangle ABC$ be a quadrantal 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 side $c$ be a right angle.
Let the remaining parts of $\triangle ABC$ be arranged according to the exterior 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 cosine of the opposite parts.
Proof
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 |
$\blacksquare$
Also see
- Napier's Cosine Rule for Right Spherical Triangles
- Napier's Tangent Rule for Right Spherical Triangles
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.