Find the y-coordinate?

Question

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.

Answer 1

http://www.wyzant.com/resources/files/263634/polar_intersection_numeric_solution

 

and

 

http://www.wyzant.com/resources/files/263648/polar_intersection_numeric_solution_2

 

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.