Relation between polar co-ordinates and three-dimensional Cartesian co-ordinates:

Let us consider, x, y, and z to be the Cartesian co-ordinates of the point A as shown in the adjoining figure.

OB is the projection of OA in the X-Y plane, since OA makes an angle  \( \theta \) with the positive Z-axis, so  \( OB=r\sin\theta \).

Since  \( \phi \) is the azimuthal angle, so

\( OC=x=OB\cos\phi=r\sin\theta\cos\phi \)

and,  \( OD=y=OB\sin\phi=r\sin\theta\sin\phi \)

and,  \( z=r\cos\theta \)

So the relation between Cartesian co-ordinates and the polar co-ordinates are as follows:

 \( x=r\sin\theta\cos\phi \)

 \( y=r\sin\theta\sin\phi \)

 \( z=r\cos\theta \)

Relation between spherical polar co-ordinates and Cartesian co-ordinates:

\(x^2+y^2+z^2=r^2{\sin}^2\theta{\cos}^2\phi+r^2{\sin}^2\theta{\sin}^2\phi+r^2{\cos}^2\theta \)

Or, \( x^2+y^2+z^2=r^2[{\sin}^2\theta+{\cos}^2\phi] \)

Or, \( x^2+y^2+z^2=r^2 \)

Or, \( r=\sqrt{ x^2+y^2+z^2} \)

Again,

\( \displaystyle{\tan\phi=\frac{r\sin\theta\sin\phi}{r\sin\theta\cos\phi}=\frac{y}{x}} \)

Or, \( \displaystyle{\phi={\tan}^{-1}\frac{y}{x}} \).

And,

\( \displaystyle{\tan\theta=\frac{r\sin\theta}{z}} \)

\( or,\ \displaystyle{\tan\theta=\frac{\sqrt{x^2+y^2}}{z}} \)