2011 AIME I Problems/Problem 9
Problem
Suppose is in the interval
and
. Find
.
Solution
We can rewrite the given expression as
.
Square both sides and divide by
to get
Rewrite
as
Testing values using the rational root theorem gives
as a root. $\Arcsin \frac{1}{3}$ (Error compiling LaTeX. Unknown error_msg) does fall in the first quadrant so it satisfies the interval. Thus
. Using the Pythagorean Identity gives us
. Then we use the definition of
24\cot ^2 x=24\frac{\cos ^2 x}{\sin ^2 x}=24\left(\frac{\frac{8}{9}}{\frac{1}{9}}\right)=24(8)=\boxed{192}$.