Delambre'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.

Then:


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


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


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


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