Delambre's Analogies/Sine by Cosine
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:
- $\sin \dfrac c 2 \cos \dfrac {A - B} 2 = \sin \dfrac C 2 \sin \dfrac {a + b} 2$
Proof
In the below, we have:
- $s = \dfrac {a + b + c} 2$
Thus:
| \(\ds \sin \frac c 2 \cos \dfrac {A - B} 2\) | \(=\) | \(\ds \sin \frac c 2 \paren {\cos \dfrac A 2 \cos \dfrac B 2 + \sin \dfrac A 2 \sin \dfrac B 2}\) | Cosine of Difference | |||||||||||
| \(\ds \) | \(=\) | \(\ds \sin \frac c 2 \paren {\sqrt {\dfrac {\sin s \map \sin {s - a} } {\sin b \sin c} } \sqrt {\dfrac {\sin s \map \sin {s - b} } {\sin a \sin c} } + \sqrt {\dfrac {\map \sin {s - b} \map \sin {s - c} } {\sin b \sin c} } \sqrt {\dfrac {\map \sin {s - a} \map \sin {s - c} } {\sin a \sin c} } }\) | Sine of Half Angle for Spherical Triangles, Cosine of Half Angle for Spherical Triangles | |||||||||||
| \(\ds \) | \(=\) | \(\ds \sin \frac c 2 \paren {\dfrac {\sin s} {\sin c} \sqrt {\dfrac {\map \sin {s - a} \map \sin {s - b} } {\sin a \sin b} } + \dfrac {\map \sin {s - c} } {\sin c} \sqrt {\dfrac {\map \sin {s - a} \map \sin {s - b} } {\sin a \sin b} } }\) | simplifying | |||||||||||
| \(\ds \) | \(=\) | \(\ds \sin \frac c 2 \sin \dfrac C 2 \paren {\dfrac {\sin s } {\sin c} + \dfrac {\map \sin {s - c} } {\sin c} }\) | Sine of Half Angle for Spherical Triangles | |||||||||||
| \(\ds \) | \(=\) | \(\ds \dfrac {\sin \frac c 2} {\sin c} \sin \dfrac C 2 \paren {\map \sin {\dfrac {a + b + c} 2} + \map \sin {\dfrac {a + b - c} 2} }\) | Definition of $s$ | |||||||||||
| \(\ds \) | \(=\) | \(\ds \dfrac {\sin \frac c 2} {\sin c} \sin \dfrac C 2 \paren {2 \map \sin {\dfrac {a + b + c + a + b - c} 4} \map \cos {\dfrac {a + b + c - a - b + c} 4} }\) | Sine plus Sine | |||||||||||
| \(\ds \) | \(=\) | \(\ds \dfrac {\sin \frac c 2} {\sin c} \sin \dfrac C 2 \paren {2 \map \sin {\dfrac {a + b} 2} \map \cos {\dfrac c 2} }\) | simplifying | |||||||||||
| \(\ds \) | \(=\) | \(\ds \dfrac {2 \sin \frac c 2 \cos {\frac c 2} } {2 \sin \frac c 2 \cos \frac c 2} \sin \dfrac C 2 \map \sin {\dfrac {a + b} 2}\) | Double Angle Formula for Sine | |||||||||||
| \(\ds \) | \(=\) | \(\ds \sin \dfrac C 2 \map \sin {\dfrac {a + b} 2}\) | 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
- 1976: W.M. Smart: Textbook on Spherical Astronomy (6th ed.) ... (previous) ... (next): Chapter $\text I$. Spherical Trigonometry: $16$. Delambre's and Napier's analogies.