Half-side formula

{{short description|Relation between the side lengths and angles of a spherical triangle}}

Image:Law-of-haversines.svg

In spherical trigonometry, the half side formula relates the angles and lengths of the sides of spherical triangles, which are triangles drawn on the surface of a sphere and so have curved sides and do not obey the formulas for plane triangles.{{citation|title=Handbook of Mathematics|title-link=Bronshtein and Semendyayev|first1=I. N.|last1=Bronshtein|first2=K. A.|last2=Semendyayev|first3=Gerhard|last3=Musiol|first4=Heiner|last4=Mühlig|publisher=Springer|year=2007|isbn=9783540721222|page=165}}[https://books.google.com/books?id=gCgOoMpluh8C&pg=PA165]

For a triangle $\backslash triangle\; ABC$ on a sphere, the half-side formula is{{citation|title=The Penguin Dictionary of Mathematics|edition=4th|first=David|last=Nelson|publisher=Penguin UK|year=2008|isbn=9780141920870|page=529|url=https://books.google.com/books?id=ud3sEeVdTIwC&pg=PT529}}.

$$\backslash begin\{align\}$$

\tan \tfrac12 a

&= \sqrt{\frac{-\cos(S)\, \cos(S - A)}

{\cos(S - B)\, \cos(S - C)} }

\end{align}

where {{mvar|a, b, c}} are the angular lengths (measure of central angle, arc lengths normalized to a sphere of unit radius) of the sides opposite angles {{mvar|A, B, C}} respectively, and $S\; =\; \backslash tfrac12\; (A+B+\; C)$ is half the sum of the angles. Two more formulas can be obtained for $b$ and $c$ by permuting the labels $A,\; B,\; C.$

The polar dual relationship for a spherical triangle is the half-angle formula,

$$\backslash begin\{align\}$$

\tan \tfrac12 A

&= \sqrt{\frac{\sin(s - b)\, \sin(s - c)}

{\sin(s)\, \sin(s - a)} }

\end{align}

where semiperimeter $s\; =\; \backslash tfrac12\; (a\; +\; b\; +\; c)$ is half the sum of the sides. Again, two more formulas can be obtained by permuting the labels $A,\; B,\; C.$