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Law of Sines

LawofSines

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/(sinA)=b/(sinB)=c/(sinC)=2R,
(1)

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

a(sinB-sinC)+b(sinC-sinA)+c(sinA-sinB)=0
(2)
a=bcosC+ccosB,
(3)

the law of cosines

cosA=(c^2+b^2-a^2)/(2bc),
(4)

and the law of tangents

(a+b)/(a-b)=(tan[1/2(A+B)])/(tan[1/2(A-B)]).
(5)

The law of sines for oblique spherical triangles states that

(sina)/(sinA)=(sinb)/(sinB)=(sinc)/(sinC).
(6)

See also

Generalized Law of Sines, Law of Cosines, Law of Tangents Explore this topic in the MathWorld classroom

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References

Abramowitz, M. and Stegun, I. 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. "The Extended Law of Sines." §1.1 in Geometry Revisited. Washington, DC: Math. Assoc. Amer., pp. 1-3, 1967.

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Law of Sines

Cite this as:

Weisstein, Eric W. "Law of Sines." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/LawofSines.html

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