Delambre's Analogies/Cosine by Cosine

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Delambre'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:

$\cos \dfrac c 2 \cos \dfrac {A + B} 2 = \sin \dfrac C 2 \cos \dfrac {a + b} 2$


Proof

In the below, we have:

$s = \dfrac {a + b + c} 2$


Thus:

\(\ds \cos \frac c 2 \cos \dfrac {A + B} 2\) \(=\) \(\ds \cos \frac c 2 \paren {\cos \dfrac A 2 \cos \dfrac B 2 - \sin \dfrac A 2 \sin \dfrac B 2}\) Cosine of Sum
\(\ds \) \(=\) \(\ds \cos \frac c 2 \paren {\sqrt {\dfrac {\sin s \, \map \sin {s - a} } {\sin b \sin c} } \sqrt {\dfrac {\sin s \, \map \sin {s - b} } {\sin a \sin c} } - \sqrt {\dfrac {\map \sin {s - b} \, \map \sin {s - c} } {\sin b \sin c} } \sqrt {\dfrac {\map \sin {s - a} \, \map \sin {s - c} } {\sin a \sin c} } }\) Sine of Half Angle for Spherical Triangles, Cosine of Half Angle for Spherical Triangles
\(\ds \) \(=\) \(\ds \cos \frac c 2 \paren {\dfrac {\sin s} {\sin c} \sqrt {\dfrac {\map \sin {s - a} \, \map \sin {s - b} } {\sin a \sin b} } - \dfrac {\map \sin {s - c} } {\sin c} \sqrt {\dfrac {\map \sin {s - a} \, \map \sin {s - b} } {\sin a \sin b} } }\) simplifying
\(\ds \) \(=\) \(\ds \cos \frac c 2 \sin \dfrac C 2 \paren {\dfrac {\sin s } {\sin c} - \dfrac {\map \sin {s - c} } {\sin c} }\) Sine of Half Angle for Spherical Triangles
\(\ds \) \(=\) \(\ds \dfrac {\cos \frac c 2} {\sin c} \sin \dfrac C 2 \paren {\map \sin {\dfrac {a + b + c} 2} - \map \sin {\dfrac {a + b - c} 2} }\) Definition of $s$
\(\ds \) \(=\) \(\ds \dfrac {\cos \frac c 2} {\sin c} \sin \dfrac C 2 \paren {2 \map \cos {\dfrac {a + b + c + a + b - c} 4} \, \map \sin {\dfrac {a + b + c - a - b + c} 4} }\) Sine minus Sine
\(\ds \) \(=\) \(\ds \dfrac {\cos \frac c 2} {\sin c} \sin \dfrac C 2 \paren {2 \map \cos {\dfrac {a + b} 2} \, \map \sin {\dfrac c 2} }\) simplifying
\(\ds \) \(=\) \(\ds \dfrac {2 \cos \frac c 2 \sin {\frac c 2} } {2 \sin \frac c 2 \cos \frac c 2} \, \sin \dfrac C 2 \, \map \cos {\dfrac {a + b} 2}\) Double Angle Formula for Sine
\(\ds \) \(=\) \(\ds \sin \dfrac C 2 \, \map \cos {\dfrac {a + b} 2}\) simplifying

$\blacksquare$


Also known as

Delambre's Analogies are also known as Gauss's Formulas, or Gauss's Formulae.

However, there are so many results and theorems named for Carl Friedrich Gauss that $\mathsf{Pr} \infty \mathsf{fWiki}$ prefers to settle for Delambre.


The names of the individual formulas are not standard, but $\mathsf{Pr} \infty \mathsf{fWiki}$ needs some way to distinguish between them. Any advice on this matter is welcome.


Also see


Source of Name

This entry was named for Jean Baptiste Joseph Delambre.


Sources