Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

2013 AMC 12B Problems/Problem 20

Revision as of 10:23, 29 January 2014 by Lightest (talk | contribs) (→‎Solution)

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 $\sin(2x)$?

$\textbf{(A)}\ 2-2\sqrt{2}\qquad\textbf{(B)}\3\sqrt{3}-6\qquad\textbf{(C)}\ 3\sqrt{2}-5\qquad\textbf{(D}}\ -\frac{3}{4}\qquad\textbf{(E)}\ 1-\sqrt{3}$ (Error compiling LaTeX. Unknown error_msg)

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

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. AMC logo.png

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=2013_AMC_12B_Problems/Problem_20&oldid=58960"