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Law of Tangents

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The law of tangents states that if a triangle has sides of length a, b, and c opposite angles A, B, and C, then

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

An analogous result for oblique spherical triangles states that

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

For a hyperbolic triangle of curvature -R^(-2), the corresponding identity is the hyperbolic law of tangents,

(tanh[1/2(a-b)/R])/(tanh[1/2(a+b)/R])=(tan[1/2(A-B)])/(tan[1/2(A+B)]).
(3)

Here, tanh is the hyperbolic tangent. This identity follows from the hyperbolic law of sines by applying sum-to-product identities (Anderson 1999, pp. 146-151).

See also

Hyperbolic Law of Sines, Hyperbolic Law of Tangents, Hyperbolic Triangle, Law of Cosines, Law of Sines, Tangent

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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, pp. 145 and 149, 1987.

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Law of Tangents

Cite this as:

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

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