Law of Tangents
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.
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Problems
Introductory
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Intermediate
In , let
be a point in
such that
bisects
. Given that
, and
, find
.
Olympiad
Show that .