Answer
$$\left( {\frac{5}{2},\frac{5}{2}, - \frac{{5\sqrt 2 }}{2}} \right)$$
Work Step by Step
$$\eqalign{
& \left( {5,\frac{\pi }{4},\frac{{3\pi }}{4}} \right) \cr
& \left( {\rho ,\theta ,\phi } \right):{\text{ }}\left( {5,\frac{\pi }{4},\frac{{3\pi }}{4}} \right) \to \rho = 5,{\text{ }}\theta = \frac{\pi }{4},{\text{ }}\phi = \frac{{3\pi }}{4} \cr
& {\text{Spherical to rectangular }}\left( {\rho ,\theta ,\phi } \right) \cr
& x = \rho \sin \phi \cos \theta ,{\text{ }}y = \rho \sin \phi \sin \theta ,{\text{ }}z = \rho \cos \phi \cr
& x = \left( 5 \right)\sin \left( {\frac{\pi }{4}} \right)\cos \left( {\frac{{3\pi }}{4}} \right) = \frac{5}{2} \cr
& y = \left( 5 \right)\sin \left( {\frac{\pi }{4}} \right)\sin \left( {\frac{{3\pi }}{4}} \right) = \frac{5}{2} \cr
& z = \left( 5 \right)\cos \left( {\frac{{3\pi }}{4}} \right) = - \frac{{5\sqrt 2 }}{2} \cr
& {\text{The rectangular }}\left( {x,y,z} \right){\text{ coordinates are:}} \cr
& \left( {\frac{5}{2},\frac{5}{2}, - \frac{{5\sqrt 2 }}{2}} \right) \cr} $$
