Napier's Analogies/Tangent of Half Sum of Angles
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Napier's Analogies
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.
Then:
- $\tan \dfrac {A + B} 2 = \dfrac {\cos \frac {a - b} 2} {\cos \frac {a + b} 2} \cot \dfrac C 2$
Proof
\(\text {(1)}: \quad\) | \(\ds \cos \dfrac c 2 \sin \dfrac {A + B} 2\) | \(=\) | \(\ds \cos \dfrac C 2 \cos \dfrac {a - b} 2\) | Delambre's Analogies | ||||||||||
\(\text {(2)}: \quad\) | \(\ds \cos \dfrac c 2 \cos \dfrac {A + B} 2\) | \(=\) | \(\ds \sin \dfrac C 2 \cos \dfrac {a + b} 2\) | Delambre's Analogies | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds \tan \dfrac {A + B} 2\) | \(=\) | \(\ds \frac {\cos \frac {a - b} 2} {\cos \frac {a + b} 2} \cot \dfrac C 2\) | dividing $(1)$ by $(2)$ |
$\blacksquare$
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