Problem 381

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Problem 381: (2013 AMC 12b, 20)

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

Exercise 381: Solution

For $0^\circ<x<90^\circ$, the points $P=(\cos x,\cos^2x)$, $Q=(\cos2x,\cos^22x)$, $R=(\cos3x,\cos^23x)$ and $S=(\cos4x,\cos^24x)$ are the vertices of a trapezoid. What is $\cos x$?

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