Difference between revisions of "Law of Tangents"
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− | The '''Law of Tangents''' is a | + | 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 [[angle]]s or sides in a [[triangle]]. |
− | == | + | == Statement == |
− | |||
− | + | 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)/2}{\tan (A+B)/2} . </cmath> | |
− | <cmath>\frac{a-b}{a+b}=\frac{\ | + | |
− | + | == Proof == | |
− | <cmath>\frac{a-b}{a+b}=\frac{\sin A-\sin B}{\sin A +\sin B}</cmath> | + | |
− | By the | + | Let <math>s</math> and <math>d</math> denote <math>(A+B)/2</math>, <math>(A-B)/2</math>, respectively. By the [[Law of Sines]], |
− | <cmath>\frac{ | + | <cmath> \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)} . </cmath> |
− | + | By the angle addition identities, | |
− | + | <cmath> \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{\tan (A-B)/2}{\tan (A+B)/2} </cmath> | |
− | as desired. | + | as desired. <math>\blacksquare</math> |
− | |||
==Problems== | ==Problems== |
Revision as of 23:04, 2 February 2008
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)