Napier's Tangent Rules for Right Angled Spherical Triangles

Napier's Rules for Right Angled Spherical Triangles: Tangents

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 tangents of the adjacent parts.

Napier's Tangent 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.

$\sin a$

\(\ds \cos a \cos C\)

\(=\)

\(\ds \sin a \cot b - \sin C \cot B\)

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

$\blacksquare$

Napier's Tangent 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.

$\sin A$

\(\ds \cos A \cos c\)

\(=\)

\(\ds \sin A \cot B - \sin c \cot b\)

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:Corollary for side $a$

\(\ds \leadsto \ \ \)

\(\ds \sin A \times 0\)

\(=\)

\(\ds \cos C \sin B + \sin C \cos B \cos a\)

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 $\sin C \cos B$

\(\ds \leadsto \ \ \)

\(\ds \map \sin {\Box - a}\)

\(=\)

\(\ds \map \tan {C - \Box} \tan b\)

Sine of Complement equals Cosine, Sine Function is Odd, Tangent of Complement equals Cotangent

$\Box$

$\map \sin {C - \Box}$

\(\ds \cos c\)

\(=\)

\(\ds \cos a \cos b + \sin a \sin b \cos C\)

Spherical Law of Cosines for side $c$

\(\ds \leadsto \ \ \)

\(\ds 0\)

\(=\)

\(\ds \cos a \cos b + \sin a \sin b \cos C\)

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 $-\sin a \sin b$

\(\ds \leadsto \ \ \)

\(\ds \map \sin {C - \Box}\)

\(=\)

\(\ds \map \tan {\Box - a} \, \map \tan {\Box - b}\)

Sine of Complement equals Cosine, Sine Function is Odd, 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:Corollary for side $b$

\(\ds \leadsto \ \ \)

\(\ds \sin B \times 0\)

\(=\)

\(\ds \cos C \sin A + \sin C \cos A \cos b\)

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 $\sin C \cos A$

\(\ds \leadsto \ \ \)

\(\ds \map \sin {\Box - b}\)

\(=\)

\(\ds \map \tan {C - \Box} \tan a\)

Sine of Complement equals Cosine, Sine Function is Odd, Tangent of Complement equals Cotangent

$\blacksquare$

Also see

• Napier's Cosine Rules for Right Angled Spherical Triangles

Source of Name

This entry was named for John Napier.