2013 AIME II Problems/Problem 15
\[ ,
,
, 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
.
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
.