Cosine of Half Angle for Spherical Triangles/Mistake

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Source Work

1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables

Chapter $5$: Trigonometrical Functions:
Relationships between Sides and Angles of a Spherical Triangle

This mistake can be seen in the edition as published by Schaum: ISBN 0-07-060224-7 (unknown printing).


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.


$\cos \dfrac A 2 = \sqrt {\dfrac {\sin s \, \map \sin {s - c} } {\sin b \sin c} }$

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


The correct formula is:

$\cos \dfrac A 2 = \sqrt {\dfrac {\sin s \, \map \sin {s - a} } {\sin b \sin c} }$