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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,
\end{displaymath} (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
\end{displaymath} (2)

a=b\cos C+c\cos B,
\end{displaymath} (3)

the Law of Cosines
\cos A={c^2+b^2-a^2\over 2bc},
\end{displaymath} (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)]}.
\end{displaymath} (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}.
\end{displaymath} (6)

See also Law of Cosines, Law of Tangents


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.

© 1996-9 Eric W. Weisstein