Using Fig.1 below, the trigonometric ratios and Pythagorean theorem, it can be shown that the relationships between

\( r = \rho \sin \phi \) , \( \theta = \theta \) , \( z = \rho \cos \phi \) (I)

\( \rho = \sqrt {r^2 + z^2} \) , \( \theta = \theta \) , \( \tan \phi = \dfrac{r}{z} \) (II)

with \( 0 \le \theta \lt 2\pi \) and \( 0 \le \phi \le \pi \) The calculator calculates the cylindrical coordinates \( r \), \( \theta \) and \( z \) given the spherical coordinates \( \rho \) , \( \theta \) and \( \phi \) using the three formulas in I.