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Hyperbolic Law of Sines

The hyperbolic law of sines relates the sides and angles of a hyperbolic triangle. For curvature -1, let a, b, and c be the side lengths opposite A, B, and C, respectively. Then

(sinha)/(sinA)=(sinhb)/(sinB)=(sinhc)/(sinC).

Here sinh is the hyperbolic sine. For curvature -R^(-2), the formula becomes

(sinh(a/R))/(sinA)=(sinh(b/R))/(sinB)=(sinh(c/R))/(sinC).

As R->infty, the curvature approaches zero and sinh(x/R)∼x/R, so the common factor 1/R cancels and the formula reduces to the Euclidean law of sines.

See also

Angle, Curvature, Hyperbolic Functions, Hyperbolic Law of Cosines, Hyperbolic Law of Tangents, Hyperbolic Plane, Hyperbolic Sine, Hyperbolic Triangle, Law of Sines, Side

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References

Anderson, J. W. "Trigonometry in the Hyperbolic Plane." §5.7 in Hyperbolic Geometry. New York: Springer-Verlag, pp. 146-151, 1999.

Cite this as:

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

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