Let us consider the Cartesian co-ordinates of the point are \( (x,\ y,\ z) \),

Given, the spherical polar co-ordinates of that point are

\( r=8 \)

\( \theta=30^{\circ} \)

\( \phi=45^{\circ} \)

We know that,

\( x=r\ \sin\theta\ \cos\phi= 8\times\sin{30^{\circ}}\times\cos{45^{\circ}} \)

\( or,\ x=8\times\frac{1}{2}\times\frac{1}{\sqrt{2}}=2\sqrt{2} \)

\( y=r\ \sin\theta\ \sin\phi=8\times\sin{30^{\circ}}\times\sin{45^{\circ}} \)

\( or,\ y=8\times\frac{1}{2}\frac{1}{\sqrt{2}}=2\sqrt{2} \)

\( z=r\ \cos\theta=8\times\cos{30^{\circ}} \)

\( or,\ z=8\times\frac{\sqrt{3}}{2}=4\sqrt{3} \)

So the Cartesian co-ordinates of the point are \( (2\sqrt{2},\ 2\sqrt{2},\ 4\sqrt{3}) \)

[ To know the relation between the three-dimensional cartesian co-ordinates and the spherical polar co-ordinates (CLICK HERE) ]