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Law of Tangents

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The Law of Tangents is a rather obscure trigonometric identity that is sometimes used in place of its better-known counterparts, the law of sines and law of cosines, to calculate angles or sides in a triangle.

Contents

Statement

If A and B are angles in a triangle opposite sides a and b respectively, then \frac{a-b}{a+b}=\frac{\tan (A-B)/2}{\tan (A+B)/2} .

Proof

Let s and d denote (A+B)/2, (A-B)/2, respectively. By the Law of Sines, \frac{a-b}{a+b} = \frac{\sin A - \sin B}{\sin A + \sin B} = \frac{ \sin(s+d) - \sin (s-d)}{\sin(s+d) + \sin(s-d)} . By the angle addition identities, \frac{\sin(s+d) - \sin(s-d)}{\sin(s+d) + \sin(s-d)} = \frac{2\cos s \sin d}{2\sin s \cos d} = \frac{\tan d}{\tan s} = \frac{\... as desired. \blacksquare

Problems

Introductory

This problem has not been edited in. If you know this problem, please help us out by adding it.

Intermediate

In \triangle ABC, let D be a point in BC such that AD bisects \angle A. Given that AD=6,BD=4, and DC=3, find AB.

Olympiad

Show that [ABC]=r^2\cot \frac{A}{2}\cot \frac{B}{2}\cot \frac{C}{2}.

(AoPS Vol. 2)

See Also

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