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Difference between revisions of "2013 AMC 12B Problems/Problem 20"

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For <math>135^\circ < x < 180^\circ</math>, points <math>P=(\cos x, \cos^2 x), Q=(\cot x, \cot^2 x), R=(\sin x, \sin^2 x)</math> and <math>S =(\tan x, \tan^2 x)</math> are the vertices of a trapezoid. What is <math>\sin(2x)</math>?
 
For <math>135^\circ < x < 180^\circ</math>, points <math>P=(\cos x, \cos^2 x), Q=(\cot x, \cot^2 x), R=(\sin x, \sin^2 x)</math> and <math>S =(\tan x, \tan^2 x)</math> are the vertices of a trapezoid. What is <math>\sin(2x)</math>?
  
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<math> \textbf{(A)}\ 2-2\sqrt{2}\qquad\textbf{(B)}\3\sqrt{3}-6\qquad\textbf{(C)}\ 3\sqrt{2}-5\qquad\textbf{(D}}\ -\frac{3}{4}\qquad\textbf{(E)}\ 1-\sqrt{3}</math>
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<math>\textbf{(A)} \ 2-2\sqrt{2}\qquad\textbf{(B)}\ 3\sqrt{3}-6\qquad\textbf{(C)}\ 3\sqrt{2}-5\qquad\textbf{(D)}\ -\frac{3}{4}\qquad\textbf{(E)}\ 1-\sqrt{3}</math>
  
 
==Solution==
 
==Solution==
  
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Let <math>f,g,h,j</math> be <math>\sin, \cos, \tan, \cot</math> (not respectively). Then we have four points <math>(f,f^2),(g,g^2),(h,h^2),(j,j^2)</math>, and the two lines connecting two pairs of them must be parallel (as we are dealing with a trapezoid). WLOG, we take the line connecting the first two and the line connecting the last two to be parallel, so that <math>\frac{g^2-f^2}{g-f} = \frac{j^2-h^2}{j-h}</math>, or <math>g+f = j+h</math>.
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Let <math>f,g,h,j</math> be <math>\sin, \cos, \tan, \cot</math> (not respectively). Then we have four points <math>(f,f^2),(g,g^2),(h,h^2),(j,j^2)</math>, and a pair of lines each connecting two points must be parallel (as we are dealing with a trapezoid). WLOG, take the line connecting the first two points and the line connecting the last two points to be parallel, so that <math>\frac{g^2-f^2}{g-f} = \frac{j^2-h^2}{j-h}</math>, or <math>g+f = j+h</math>.
  
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Now, we must find how to match up <math>\sin, \cos, \tan, \cot</math> to <math>f,g,h,j</math> so that the above equation has a solution. The three possible ways are <math>\sin x+\cos x = \tan x+\cot x</math>, <math>\sin x+\tan x = \cos x+\cot x</math>, and <math>\sin x+\cot x = \cos x+\tan x</math>. On the interval <math>135^\circ < x < 180^\circ</math>, we have <math>0<\sin x<\frac{1}{\sqrt{2}},-1< \cos x<-\frac{1}{\sqrt{2}},-1< \tan x<0,-\infty< \cot x<-1</math>, so the first two of those possible ways do not work because the LHS and the RHS have different ranges. Thus, it must be the last possible way, <math>\sin x+\cot x = \cos x+\tan x</math>.
+
Now, we must find how to match up <math>\sin, \cos, \tan, \cot</math> to <math>f,g,h,j</math> so that the above equation has a solution. On the interval <math>135^\circ < x < 180^\circ</math>, we have <math>\cot x <-1<\cos x<0<\sin x</math>, and <math>\cot x <-1<\tan x<0<\sin x</math> so the sum of the largest and the smallest is equal to the sum of the other two, namely, <math>\sin x+\cot x = \cos x+\tan x</math>.
  
 
Now, we perform some algebraic manipulation to find <math>\sin (2x)</math>:
 
Now, we perform some algebraic manipulation to find <math>\sin (2x)</math>:
  
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<cmath>
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<math>
 
\sin x+\cot x = \cos x+\tan x \\
 
\sin x+\cot x = \cos x+\tan x \\
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(\sin x+\cot x)(\sin x \cos x ) = (\cos x+\tan x)(\sin x \cos x ) \\
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\sin x - \cos x = \tan x - \cot x = (\sin x - \cos x) (\sin x + \cos x) / (\sin x \cos x) \\
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\sin^2 x \cos x+\cos^2 x = \cos^2 x \sin x+\sin^2 x \\
 
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\sin^2 x \cos x -\cos^2 x \sin x = \sin^2 x - \cos^2 x \\
 
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\sin x \cos x (\sin x -\cos x) = (\sin x - \cos x)(\sin x + \cos x) \\
 
 
\sin x \cos x = \sin x + \cos x \\
 
\sin x \cos x = \sin x + \cos x \\
 
(\sin x \cos x)^2 = (\sin x + \cos x)^2 \\
 
(\sin x \cos x)^2 = (\sin x + \cos x)^2 \\
 
(\sin x \cos x)^2 = \sin^2 x + 2\sin x \cos x + \cos^2 x \\
 
(\sin x \cos x)^2 = \sin^2 x + 2\sin x \cos x + \cos^2 x \\
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(\sin x \cos x)^2 = 1 + 2\sin x \cos x \\
 
 
(\sin x \cos x)^2 - 2\sin x \cos x - 1 =0
 
(\sin x \cos x)^2 - 2\sin x \cos x - 1 =0
−
</cmath>
+
</math>
  
 
Solve the quadratic to find <math>\sin x \cos x = \frac{2 - 2\sqrt{2}}{2}</math>, so that <math>\sin(2x) = 2 \sin x \cos x = \boxed{\textbf{(A)} \ 2 - 2\sqrt{2}}</math>.
 
Solve the quadratic to find <math>\sin x \cos x = \frac{2 - 2\sqrt{2}}{2}</math>, so that <math>\sin(2x) = 2 \sin x \cos x = \boxed{\textbf{(A)} \ 2 - 2\sqrt{2}}</math>.
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== See also ==
 
== See also ==
 
{{AMC12 box|year=2013|ab=B|num-b=19|num-a=21}}
 
{{AMC12 box|year=2013|ab=B|num-b=19|num-a=21}}
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[[Category:Introductory Geometry Problems]]
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{{MAA Notice}}

Latest revision as of 23:54, 27 September 2021

Problem

For $135^\circ < x < 180^\circ$, points $P=(\cos x, \cos^2 x), Q=(\cot x, \cot^2 x), R=(\sin x, \sin^2 x)$ and $S =(\tan x, \tan^2 x)$ are the vertices of a trapezoid. What is $\sin(2x)$?

$\textbf{(A)} \ 2-2\sqrt{2}\qquad\textbf{(B)}\ 3\sqrt{3}-6\qquad\textbf{(C)}\ 3\sqrt{2}-5\qquad\textbf{(D)}\ -\frac{3}{4}\qquad\textbf{(E)}\ 1-\sqrt{3}$

Solution

Let $f,g,h,j$ be $\sin, \cos, \tan, \cot$ (not respectively). Then we have four points $(f,f^2),(g,g^2),(h,h^2),(j,j^2)$, and a pair of lines each connecting two points must be parallel (as we are dealing with a trapezoid). WLOG, take the line connecting the first two points and the line connecting the last two points to be parallel, so that $\frac{g^2-f^2}{g-f} = \frac{j^2-h^2}{j-h}$, or $g+f = j+h$.

Now, we must find how to match up $\sin, \cos, \tan, \cot$ to $f,g,h,j$ so that the above equation has a solution. On the interval $135^\circ < x < 180^\circ$, we have $\cot x <-1<\cos x<0<\sin x$, and $\cot x <-1<\tan x<0<\sin x$ so the sum of the largest and the smallest is equal to the sum of the other two, namely, $\sin x+\cot x = \cos x+\tan x$.

Now, we perform some algebraic manipulation to find $\sin (2x)$:

$\sin x+\cot x = \cos x+\tan x \\ \sin x - \cos x = \tan x - \cot x = (\sin x - \cos x) (\sin x + \cos x) / (\sin x \cos x) \\ \sin x \cos x = \sin x + \cos x \\ (\sin x \cos x)^2 = (\sin x + \cos x)^2 \\ (\sin x \cos x)^2 = \sin^2 x + 2\sin x \cos x + \cos^2 x \\ (\sin x \cos x)^2 - 2\sin x \cos x - 1 =0$

Solve the quadratic to find $\sin x \cos x = \frac{2 - 2\sqrt{2}}{2}$, so that $\sin(2x) = 2 \sin x \cos x = \boxed{\textbf{(A)} \ 2 - 2\sqrt{2}}$.

See also

2013 AMC 12B (Problems • Answer Key • Resources)
Preceded by
Problem 19 		Followed by
Problem 21
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25
All AMC 12 Problems and Solutions

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