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Law of Sines

LawofSines

The law of sines states that for a triangle with sides of lengths a, b, and c opposite angles A, B, and C, respectively,

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)

For a triangle in the hyperbolic plane, the corresponding result is the hyperbolic law of sines. It reduces to the Euclidean formula above in the zero-curvature limit (Anderson 1999).

See also

Angle, Generalized Law of Sines, Hyperbolic Law of Sines, Law of Cosines, Law of Tangents, Side, Spherical Triangle, Triangle 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.Anderson, J. W. "Trigonometry in the Hyperbolic Plane." §5.7 in Hyperbolic Geometry. New York: Springer-Verlag, pp. 146-151, 1999.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 Resource. https://mathworld.wolfram.com/LawofSines.html

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