Difference between revisions of "Law of Tangents"
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===Intermediate=== | ===Intermediate=== | ||
− | In <math>\triangle ABC</math>, | + | In <math>\triangle ABC</math>, let <math>D</math> be a point in <math>BC</math> such that <math>AD</math> bisects <math>\angle A</math>. Given that <math>AD=6,BD=4</math>, and <math>DC=3</math>, find <math>AB</math>. |
<div align="right">([[Mu Alpha Theta]] 1991)</div> | <div align="right">([[Mu Alpha Theta]] 1991)</div> | ||
===Olympiad=== | ===Olympiad=== |
Revision as of 13:11, 2 February 2008
The Law of Tangents is a useful trigonometric identity that, along with the law of sines and law of cosines, can be used to determine angles in a triangle. Note that the law of tangents usually cannot determine sides, since only angles are involved in its statement.
Theorem
The law of tangents states that in triangle , if
and
are angles of the triangle opposite sides
and
respectively, then
.
Proof
First we can write the RHS in terms of sines and cosines:
We can use various sum-of-angle trigonometric identities to get:
By the law of sines, we have
where
is the circumradius of the triangle. Applying the law of sines again,
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
.
Olympiad
Show that .