Napier's Cosine Rules for Right Angled Spherical Triangles
Napier's Rules for Right Angled Spherical Triangles: Cosines
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 either angle $\angle C$ or side $c$ be a right angle.
Let the remaining parts of $\triangle ABC$ be arranged in a circle as above:
- for $\angle C$ a right angle, the interior
- for $c$ a right angle, the exterior
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.
The sine of the middle part equals the product of the cosine of the opposite parts.
Napier's Cosine Rule for Right Spherical Triangles
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 |
$\blacksquare$
Napier's Cosine Rule for Quadrantal Triangles
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
Source of Name
This entry was named for John Napier.