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Gauss's Formulas

Let a Spherical Triangle have sides $a$, $b$, and $c$ with $A$, $B$, and $C$ the corresponding opposite angles. Then

$\displaystyle {\sin[{\textstyle{1\over 2}}(a-b)]\over\sin({\textstyle{1\over 2}}c)}$ $\textstyle =$ $\displaystyle {\sin[{\textstyle{1\over 2}}(A-B)]\over\cos({\textstyle{1\over 2}}C)}$ (1)
$\displaystyle {\sin[{\textstyle{1\over 2}}(a+b)]\over\sin({\textstyle{1\over 2}}c)}$ $\textstyle =$ $\displaystyle {\cos[{\textstyle{1\over 2}}(A-B)]\over\sin({\textstyle{1\over 2}}C)}$ (2)
$\displaystyle {\cos[{\textstyle{1\over 2}}(a-b)]\over\cos({\textstyle{1\over 2}}c)}$ $\textstyle =$ $\displaystyle {\sin[{\textstyle{1\over 2}}(A+B)]\over\cos({\textstyle{1\over 2}}C)}$ (3)
$\displaystyle {\cos[{\textstyle{1\over 2}}(a+b)]\over\cos({\textstyle{1\over 2}}c)}$ $\textstyle =$ $\displaystyle {\cos[{\textstyle{1\over 2}}(A+B)]\over\sin({\textstyle{1\over 2}}C)}.$ (4)

See also Spherical Trigonometry




© 1996-9 Eric W. Weisstein
1999-05-25