Answer
$$\left( {\frac{{5\sqrt 2 }}{2}, - \frac{{5\sqrt 2 }}{2}} \right)$$
Work Step by Step
$$\eqalign{
& {\text{ Let the polar coordinates }}\left( {5,315^\circ } \right){\text{ }} \cr
& r = 5{\text{ and }}\theta = 315^\circ \cr
& \cr
& {\text{Convert to rectangular coordinates }}\left( {x,y} \right) \cr
& x = r\cos \theta \cr
& x = 5\cos \left( {315^\circ } \right) \cr
& x = 5\left( {\frac{{\sqrt 2 }}{2}} \right) = \frac{{5\sqrt 2 }}{2} \cr
& and \cr
& y = r\sin \theta \cr
& y = 5\sin \left( {315^\circ } \right) \cr
& y = 5\left( { - \frac{{\sqrt 2 }}{2}} \right) = - \frac{{5\sqrt 2 }}{2} \cr
& {\text{The rectangular coordinates are:}} \cr
& \left( {\frac{{5\sqrt 2 }}{2}, - \frac{{5\sqrt 2 }}{2}} \right) \cr} $$
