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Find the y-coordinate?

On the interval [0, 2pi] there is one point on the curve r=θ-2cosθ whose x-coordinate is 2. Find the y-coordinate there. 
 
Answer: -3.764
 
Please show all your work.

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Steve S. | Tutoring in Precalculus, Trig, and Differential CalculusTutoring in Precalculus, Trig, and Diffe...
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and
 
 
approximate the solution by finding the intersections of two graphs. [Using Mac OS X's Grapher utility and GeoGebra.]
 
I'm stumped on how to do this analytically.