Difference between revisions of "2000 AIME II Problems/Problem 15"

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Find the least positive integer <math>n</math> such that <center><math>\frac 1{\sin 45^\circ\sin 46^\circ}+\frac 1{\sin 47^\circ\sin 48^\circ}+\cdots+\frac 1{\sin 133^\circ\sin 134^\circ}=\frac 1{\sin n^\circ}.</math></center>
 
Find the least positive integer <math>n</math> such that <center><math>\frac 1{\sin 45^\circ\sin 46^\circ}+\frac 1{\sin 47^\circ\sin 48^\circ}+\cdots+\frac 1{\sin 133^\circ\sin 134^\circ}=\frac 1{\sin n^\circ}.</math></center>
  
== Solution ==
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== Solution 1 ==
{{solution}}
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We apply the identity
  
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<cmath>\begin{align*}
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\frac{1}{\sin n \sin (n+1)} &= \frac{1}{\sin 1} \cdot \frac{\sin (n+1) \cos n - \sin n \cos (n+1)}{\sin n \sin (n+1)} \\ &= \frac{1}{\sin 1} \cdot \left(\frac{\cos n}{\sin n} - \frac{\cos (n+1)}{\sin (n+1)}\right) \\ &= \frac{1}{\sin 1} \cdot \left(\cot n - \cot (n+1)\right). \end{align*}</cmath>
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The motivation for this identity arises from the need to decompose those fractions, possibly into [[telescoping]].
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Thus our summation becomes
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<cmath>\sum_{k=23}^{67} \frac{1}{\sin (2k-1) \sin 2k} = \frac{1}{\sin 1} \left(\cot 45 - \cot 46 + \cot 47 - \cdots + \cot 133 - \cot 134 \right).</cmath>
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Since <math>\cot (180 - x) = - \cot x</math>, the summation simply reduces to <math>\frac{1}{\sin 1} \cdot \left( \cot 45 - \cot 90 \right) = \frac{1 - 0}{\sin 1} = \frac{1}{\sin 1^{\circ}}</math>. Therefore, the answer is <math>\boxed{001}</math>.
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== Solution 2 ==
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We can make an approximation by observing the following points:
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The average term is around the 60's which gives <math>\frac{4}{3}</math>.
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There are 45 terms, so the approximate sum is 60.
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Therefore, the entire thing equals approximately <math>\frac{1}{60}</math>.
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Recall that the approximation of <math>\sin(x)</math> in radians is x if x is close to zero. In this case x is close to zero. Converting to radians we see that <math>\sin(1)</math> in degrees is about sin<math>\frac{1}{57}</math> in radians, or is about <math>\frac{1}{57}</math> because of the approximation. What we want is apparently close to that so we make the guess that n is equal to 1 degree. Basically, it boils down to the approximation of <math>\sin(1)=\frac{1}{60}</math> in degrees, convert to radians and use the small angle approximation <math>\sin(x)=x</math>.
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== See also ==
 
{{AIME box|year=2000|n=II|num-b=14|after=Last Question}}
 
{{AIME box|year=2000|n=II|num-b=14|after=Last Question}}
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[[Category:Intermediate Trigonometry Problems]]
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{{MAA Notice}}

Latest revision as of 09:37, 26 October 2016

Problem

Find the least positive integer $n$ such that

$\frac 1{\sin 45^\circ\sin 46^\circ}+\frac 1{\sin 47^\circ\sin 48^\circ}+\cdots+\frac 1{\sin 133^\circ\sin 134^\circ}=\frac 1{\sin n^\circ}.$

Solution 1

We apply the identity

\begin{align*} \frac{1}{\sin n \sin (n+1)} &= \frac{1}{\sin 1} \cdot \frac{\sin (n+1) \cos n - \sin n \cos (n+1)}{\sin n \sin (n+1)} \\ &= \frac{1}{\sin 1} \cdot \left(\frac{\cos n}{\sin n} - \frac{\cos (n+1)}{\sin (n+1)}\right) \\ &= \frac{1}{\sin 1} \cdot \left(\cot n - \cot (n+1)\right). \end{align*}

The motivation for this identity arises from the need to decompose those fractions, possibly into telescoping.

Thus our summation becomes

\[\sum_{k=23}^{67} \frac{1}{\sin (2k-1) \sin 2k} = \frac{1}{\sin 1} \left(\cot 45 - \cot 46 + \cot 47 - \cdots + \cot 133 - \cot 134 \right).\]

Since $\cot (180 - x) = - \cot x$, the summation simply reduces to $\frac{1}{\sin 1} \cdot \left( \cot 45 - \cot 90 \right) = \frac{1 - 0}{\sin 1} = \frac{1}{\sin 1^{\circ}}$. Therefore, the answer is $\boxed{001}$.

Solution 2

We can make an approximation by observing the following points:

The average term is around the 60's which gives $\frac{4}{3}$.

There are 45 terms, so the approximate sum is 60.

Therefore, the entire thing equals approximately $\frac{1}{60}$.

Recall that the approximation of $\sin(x)$ in radians is x if x is close to zero. In this case x is close to zero. Converting to radians we see that $\sin(1)$ in degrees is about sin$\frac{1}{57}$ in radians, or is about $\frac{1}{57}$ because of the approximation. What we want is apparently close to that so we make the guess that n is equal to 1 degree. Basically, it boils down to the approximation of $\sin(1)=\frac{1}{60}$ in degrees, convert to radians and use the small angle approximation $\sin(x)=x$.

See also

2000 AIME II (ProblemsAnswer KeyResources)
Preceded by
Problem 14
Followed by
Last Question
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
All AIME Problems and Solutions

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