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Difference between revisions of "2011 AIME I Problems/Problem 9"

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<cmath>24\sin ^3 x=1-\sin ^2 x</cmath>
 
<cmath>24\sin ^3 x=1-\sin ^2 x</cmath>
 
<cmath>24\sin ^3 x+\sin ^2 x - 1=0</cmath>
 
<cmath>24\sin ^3 x+\sin ^2 x - 1=0</cmath>
−
Testing values using the rational root theorem gives <math>\sin x=\frac{1}{3}</math> as a root. <math>\Arcsin \frac{1}{3}</math> does fall in the first quadrant so it satisfies the interval. Thus  
+
Testing values using the rational root theorem gives <math>\sin x=\frac{1}{3}</math> as a root. <math>\sin^{-1} \frac{1}{3}</math> does fall in the first quadrant so it satisfies the interval. Thus  
 
<cmath>\sin ^2 x=\frac{1}{9}</cmath>
 
<cmath>\sin ^2 x=\frac{1}{9}</cmath>
 
Using the Pythagorean Identity gives us <math>\cos ^2 x=\frac{8}{9}</math>. Then we use the definition of <math>\cot ^2 x</math> to compute our final answer. <math>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}</math>.
 
Using the Pythagorean Identity gives us <math>\cos ^2 x=\frac{8}{9}</math>. Then we use the definition of <math>\cot ^2 x</math> to compute our final answer. <math>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}</math>.

Revision as of 11:53, 19 March 2011

Problem

Suppose $x$ is in the interval $[0, \pi/2]$ and $\log_{24\sin x} (24\cos x)=\frac{3}{2}$. Find $24\cot^2 x$.

Solution

We can rewrite the given expression as \[\sqrt{24^3\sin^3 x}=24\cos x\] Square both sides and divide by $24^2$ to get \[24\sin ^3 x=\cos ^2 x\] Rewrite $\cos ^2 x$ as $1-\sin ^2 x$ \[24\sin ^3 x=1-\sin ^2 x\] \[24\sin ^3 x+\sin ^2 x - 1=0\] Testing values using the rational root theorem gives $\sin x=\frac{1}{3}$ as a root. $\sin^{-1} \frac{1}{3}$ does fall in the first quadrant so it satisfies the interval. Thus \[\sin ^2 x=\frac{1}{9}\] Using the Pythagorean Identity gives us $\cos ^2 x=\frac{8}{9}$. Then we use the definition of $\cot ^2 x$ to compute our final answer. $24\cot ^2 x=24\frac{\cos ^2 x}{\sin ^2 x}=24\left(\frac{\frac{8}{9}}{\frac{1}{9}}\right)=24(8)=\boxed{192}$.

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