Newton's Formulas

Let a Triangle have side lengths $a$, $b$, and $c$ with opposite angles $A$, $B$, and $C$. Then

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

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

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

See also Mollweide's Formulas, Triangle

References

Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 146, 1987.