Gauss's Formulas

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

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

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

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

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

See also Spherical Trigonometry