Difference between revisions of "2002 AIME II Problems/Problem 10"

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While finding the sine of a certain angle, an absent-minded professor failed to notice that his calculator was not in the correct angular mode. He was lucky to get the right answer. The two least positive real values of <math>x</math> for which the sine of <math>x</math> degrees is the same as the sine of <math>x</math> radians are <math>\frac{m\pi}{n-\pi}</math> and <math>\frac{p\pi}{q+\pi}</math>, where <math>m</math>, <math>n</math>, <math>p</math>, and <math>q</math> are positive integers. Find <math>m+n+p+q</math>.
 
While finding the sine of a certain angle, an absent-minded professor failed to notice that his calculator was not in the correct angular mode. He was lucky to get the right answer. The two least positive real values of <math>x</math> for which the sine of <math>x</math> degrees is the same as the sine of <math>x</math> radians are <math>\frac{m\pi}{n-\pi}</math> and <math>\frac{p\pi}{q+\pi}</math>, where <math>m</math>, <math>n</math>, <math>p</math>, and <math>q</math> are positive integers. Find <math>m+n+p+q</math>.
  
== Solution ==
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== Solution 1==
 
Note that <math>x</math> degrees is equal to <math>\frac{\pi x}{180}</math> radians. Also, for <math>\alpha \in \left[0 , \frac{\pi}{2} \right]</math>, the two least positive angles <math>\theta > \alpha</math> such that <math>\sin{\theta} = \sin{\alpha}</math> are <math>\theta = \pi-\alpha</math>, and <math>\theta = 2\pi + \alpha</math>.  
 
Note that <math>x</math> degrees is equal to <math>\frac{\pi x}{180}</math> radians. Also, for <math>\alpha \in \left[0 , \frac{\pi}{2} \right]</math>, the two least positive angles <math>\theta > \alpha</math> such that <math>\sin{\theta} = \sin{\alpha}</math> are <math>\theta = \pi-\alpha</math>, and <math>\theta = 2\pi + \alpha</math>.  
  
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<math>\theta = 2\pi + \alpha</math> yields: <math>x = 2\pi + \frac{\pi x}{180} \Rightarrow x = \frac{360\pi}{180-\pi} \Rightarrow (m,n) = (360,180)</math>.  
 
<math>\theta = 2\pi + \alpha</math> yields: <math>x = 2\pi + \frac{\pi x}{180} \Rightarrow x = \frac{360\pi}{180-\pi} \Rightarrow (m,n) = (360,180)</math>.  
  
So, <math>m+n+p+q = \boxed{900}</math>.  
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So, <math>m+n+p+q = \boxed{900}</math>.
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== Solution 2 ==
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The first case is when the two angles, <math>x</math> and <math>\frac{\pi x}{180}</math>, are coterminal. The second case is when they are reflections of the <math>y</math> axis.
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1. <math>x+2\pi a = \frac{\pi x}{180}</math> for any integer <math>a</math>
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So <math>x=\frac{360\pi a }{\pi -180}</math>
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2. <math>(2b+1)\pi -x = \frac{\pi x}{180}</math> for any integer <math>b</math>
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So <math>x = \frac{180(2b+1)\pi}{\pi + 180}</math>
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Choosing carefully <math>a,b</math> such that it's the minimum gives the answer same as above.
  
 
== See also ==
 
== See also ==

Revision as of 18:30, 24 June 2020

Problem

While finding the sine of a certain angle, an absent-minded professor failed to notice that his calculator was not in the correct angular mode. He was lucky to get the right answer. The two least positive real values of $x$ for which the sine of $x$ degrees is the same as the sine of $x$ radians are $\frac{m\pi}{n-\pi}$ and $\frac{p\pi}{q+\pi}$, where $m$, $n$, $p$, and $q$ are positive integers. Find $m+n+p+q$.

Solution 1

Note that $x$ degrees is equal to $\frac{\pi x}{180}$ radians. Also, for $\alpha \in \left[0 , \frac{\pi}{2} \right]$, the two least positive angles $\theta > \alpha$ such that $\sin{\theta} = \sin{\alpha}$ are $\theta = \pi-\alpha$, and $\theta = 2\pi + \alpha$.

Clearly $x > \frac{\pi x}{180}$ for positive real values of $x$.

$\theta = \pi-\alpha$ yields: $x = \pi - \frac{\pi x}{180} \Rightarrow x = \frac{180\pi}{180+\pi} \Rightarrow (p,q) = (180,180)$.

$\theta = 2\pi + \alpha$ yields: $x = 2\pi + \frac{\pi x}{180} \Rightarrow x = \frac{360\pi}{180-\pi} \Rightarrow (m,n) = (360,180)$.

So, $m+n+p+q = \boxed{900}$.

Solution 2

The first case is when the two angles, $x$ and $\frac{\pi x}{180}$, are coterminal. The second case is when they are reflections of the $y$ axis. 1. $x+2\pi a = \frac{\pi x}{180}$ for any integer $a$ So $x=\frac{360\pi a }{\pi -180}$

2. $(2b+1)\pi -x = \frac{\pi x}{180}$ for any integer $b$ So $x = \frac{180(2b+1)\pi}{\pi + 180}$

Choosing carefully $a,b$ such that it's the minimum gives the answer same as above.

See also

2002 AIME II (ProblemsAnswer KeyResources)
Preceded by
Problem 9
Followed by
Problem 11
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
All AIME Problems and Solutions

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