Napier's Analogies
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
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
- 1976: W.M. Smart: Textbook on Spherical Astronomy (6th ed.) ... (previous) ... (next): Chapter $\text I$. Spherical Trigonometry: $16$. Delambre's and Napier's analogies.
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): Napier's analogies
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): Napier's analogies