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. ∎
Problems
Introductory
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Intermediate
In , LET BE A POINT IN SUCH THAT bisects . Given that , and , find .
Olympiad
Show that .