Answer
$${r^2} = 4$$
Work Step by Step
$$\eqalign{
& {x^2} + {y^2} = 4 \cr
& {\text{Use }}r\sin \theta = y{\text{ and }}r\cos \theta = x,{\text{ then}} \cr
& {\left( {r\cos \theta } \right)^2} + {\left( {r\sin \theta } \right)^2} = 4 \cr
& {r^2}{\cos ^2}\theta + {r^2}{\sin ^2}\theta = 4 \cr
& {\text{factor and simplify}} \cr
& {r^2}\left( {{{\cos }^2}\theta + {{\sin }^2}\theta } \right) = 4 \cr
& {r^2}\left( 1 \right) = 4 \cr
& {r^2} = 4 \cr} $$
