Difference between revisions of "2011 AIME I Problems/Problem 9"
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== Problem == | == Problem == | ||
− | Suppose <math>x</math> is in the interval <math>[0, \pi/2]</math> and <math>\log_ | + | Suppose <math>x</math> is in the interval <math>[0, \pi/2]</math> and <math>\log_{24\sin x} (24\cos x)=\frac{3}{2}</math>. Find <math>24\cot^2 x</math>. |
== Solution == | == Solution == |
Revision as of 12:50, 19 March 2011
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
to compute our final answer.
.