Law of Sines

Let $a$, $b$, and $c$ be the lengths of the Legs of a Triangle opposite Angles $A$, $B$, and $C$. Then the law of sines states that

$${a\over\sin A}={b\over\sin B}={c\over\sin C}=2R,$$ (1)

where $R$ is the radius of the Circumcircle. Other related results include the identities

$$a(\sin B-\sin C)+b(\sin C-\sin A)+c(\sin A-\sin B)=0$$ (2)

$$a=b\cos C+c\cos B,$$ (3)

the Law of Cosines

$$\cos A={c^2+b^2-a^2\over 2bc},$$ (4)

and the Law of Tangents

$${a+b\over a-b} = {\tan[{\textstyle{1\over 2}}(A+B)]\over \tan[{\textstyle{1\over 2}}(A-B)]}.$$ (5)

The law of sines for oblique Spherical Triangles states that

$${\sin a\over\sin A}={\sin b\over\sin B}={\sin c\over\sin C}.$$ (6)

See also Law of Cosines, Law of Tangents

References

Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 79, 1972.

Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 148, 1987.

Coxeter, H. S. M. and Greitzer, S. L. Geometry Revisited. Washington, DC: Math. Assoc. Amer., pp. 1-3, 1967.