Napier's Analogies

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Theorem

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.


Tangent of Half Sum of Sides

$\tan \dfrac {a + b} 2 = \dfrac {\cos \frac {A - B} 2} {\cos \frac {A + B} 2} \tan \dfrac c 2$


Tangent of Half Difference of Sides

$\tan \dfrac {a - b} 2 = \dfrac {\sin \frac {A - B} 2} {\sin \frac {A + B} 2} \tan \dfrac c 2$


Tangent of Half Sum of Angles

$\tan \dfrac {A + B} 2 = \dfrac {\cos \frac {a - b} 2} {\cos \frac {a + b} 2} \cot \dfrac C 2$


Tangent of Half Difference of Angles

$\tan \dfrac {A - B} 2 = \dfrac {\sin \frac {a - b} 2} {\sin \frac {a + b} 2} \cot \dfrac C 2$


Also presented as

Napier's Analogies can also be seen presented as:

\(\text {(1)}: \quad\) \(\ds \dfrac {\tan \frac {a + b} 2} {\tan \frac c 2}\) \(=\) \(\ds \dfrac {\cos \frac {A - B} 2} {\cos \frac {A + B} 2}\)
\(\text {(2)}: \quad\) \(\ds \dfrac {\tan \frac {a - b} 2} {\tan \frac c 2}\) \(=\) \(\ds \dfrac {\sin \frac {A - B} 2} {\sin \frac {A + B} 2}\)
\(\text {(3)}: \quad\) \(\ds \dfrac {\tan \frac {A + B} 2} {\cot \frac C 2}\) \(=\) \(\ds \dfrac {\cos \frac {a - b} 2} {\cos \frac {a + b} 2}\)
\(\text {(4)}: \quad\) \(\ds \dfrac {\tan \frac {A - B} 2} {\cot \frac C 2}\) \(=\) \(\ds \dfrac {\sin \frac {a - b} 2} {\sin \frac {a + b} 2}\)


It is supposed that they could also be presented as:

\(\text {(1)}: \quad\) \(\ds \tan \frac {a + b} 2 \cos \frac {A + B} 2\) \(=\) \(\ds \cos \frac {A - B} 2 \tan \frac c 2\)
\(\text {(2)}: \quad\) \(\ds \tan \frac {a - b} 2 \sin \frac {A + B} 2\) \(=\) \(\ds \sin \frac {A - B} 2 \tan \frac c 2\)
\(\text {(3)}: \quad\) \(\ds \tan \frac {A + B} 2 \cos \frac {a + b} 2\) \(=\) \(\ds \cos \frac {a - b} 2 \cot \frac C 2\)
\(\text {(4)}: \quad\) \(\ds \tan \frac {A - B} 2 \sin \frac {a + b} 2\) \(=\) \(\ds \sin \frac {a - b} 2 \cot \frac C 2\)

but it has not been established that this appears anywhere in the literature.


Also see


Source of Name

This entry was named for John Napier.


Sources