2013 AIME II Problems/Problem 15
Revision as of 13:41, 4 April 2013 by ProbaBillity (talk | contribs) (I am posting Problem 15 (Problem LaTeXed by v_Enhance on the AoPS Forum) along with my solution.)
\[ ,
,
, and
for which
where
and
are relatively prime and
is not divisible by the square of any prime. Find
.
Solution
Let's draw the triangle. Since the problem only deals with angles, we can go ahead and set one of the sides to a convenient value. Let .
By the Law of Sines, we must have and
.
[geogebra]e0fd25e3d64e320f9fb1773bb77403f8b516e67e[/geogebra]
Now let us analyze the given:
$
Therefore: Similarly,
Note that the desired value is equivalent to
, which is
. All that remains is to use the sine addition formula and, after a few minor computations, we obtain a result of
. Thus, the answer is
.