# Difference between revisions of "Law of Tangents"

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If <math>A</math> and <math>B</math> are angles in a triangle opposite sides <math>a</math> and <math>b</math> respectively, then | If <math>A</math> and <math>B</math> are angles in a triangle opposite sides <math>a</math> and <math>b</math> respectively, then | ||

− | <cmath> \frac{a-b}{a+b}=\frac{\tan (A-B) | + | <cmath> \frac{a-b}{a+b}=\frac{\tan [\frac{1}{2}(A-B)]}{\tan [\frac{1}{2}(A+B)]} . </cmath> |

== Proof == | == Proof == |

## Revision as of 22:23, 25 October 2013

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.

## Statement

If and are angles in a triangle opposite sides and respectively, then

## Proof

Let and denote , , respectively. By the Law of Sines, By the angle addition identities, as desired.

## Problems

### Introductory

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

### Intermediate

In , let be a point in such that bisects . Given that , and , find .

(Mu Alpha Theta 1991)

### Olympiad

Show that .

(AoPS Vol. 2)