Answer
$(5,3)$
Work Step by Step
Given: $(x,y)=(\sqrt 2,4\sqrt 2)$ and $\phi =45^{\circ}$
The polar equation for x-component along X-axis is given as:
$X=x \cos \phi+y\sin \phi=(\sqrt 2) \cos 45^{\circ} +(4\sqrt 2)\sin 45^{\circ}\\=(\sqrt 2)\dfrac{1}{\sqrt 2}+(4\sqrt 2)\dfrac{1}{\sqrt 2}\\=5$
The polar equation for y-component along Y-axis is given as:
$Y=-x \sin \phi+y\cos \phi=-(\sqrt 2) \sin 45^{\circ}+(4\sqrt 2)\cos 45^{\circ}\\=-(\sqrt 2)\dfrac{1}{\sqrt 2}+(4\sqrt 2)\dfrac{1}{\sqrt 2}\\=3$
Thus, $(X,Y)=(5,3)$
