Answer
$$\left( {\sqrt 2 , - \sqrt 2 } \right)$$
Work Step by Step
$$\eqalign{
& \left( {2,\frac{{7\pi }}{4}} \right) \Rightarrow r = 2{\text{ and }}\theta = \frac{{7\pi }}{4} \cr
& {\text{Converting to cartesian coordinates }}\left( {x,y} \right).{\text{ Where}} \cr
& x = r\cos \theta {\text{ and }}y = r\sin \theta \cr
& {\text{We obtain}}{\text{,}} \cr
& x = 2\cos \left( {\frac{{7\pi }}{4}} \right) = 2\left( {\frac{{\sqrt 2 }}{2}} \right) = \sqrt 2 \cr
& y = 2\sin \left( {\frac{{7\pi }}{4}} \right) = 2\left( { - \frac{{\sqrt 2 }}{2}} \right) = - \sqrt 2 \cr
& {\text{The cartesian coordinates are:}} \cr
& \left( {x,y} \right) = \left( {\sqrt 2 , - \sqrt 2 } \right) \cr} $$
