Answer
a) $(2, 0,\dfrac{\pi}{6})$
b) $(4, \dfrac{11\pi}{6},\dfrac{\pi}{6})$
Work Step by Step
Conversion of rectangular to spherical coordinates is as follows:
$x=\rho \sin \phi \cos \theta; y=\rho \sin \phi \sin \theta;z=\rho \cos \phi$
and $\rho=\sqrt {x^2+y^2+z^2}$; $\cos \phi =\dfrac{z}{\rho}$; $\cos \theta=\dfrac{x}{\rho \sin \phi}$
a) Here, $\rho=2$
$\cos \phi =\dfrac{\sqrt 3}{2} \implies \phi=\dfrac{\pi}{6}$; $\cos \theta=\dfrac{0}{2 \sin \dfrac{\pi}{6}} \implies \theta=0$
Thus, $(r, \theta, \phi)=(2, 0,\dfrac{\pi}{6})$
b) Here, $\rho=4$
$\cos \phi =\dfrac{\sqrt 3}{2} \implies \phi=\dfrac{ \pi}{6}$; $\cos \theta=\dfrac{-1}{2 \sin \dfrac{\pi}{6}} \implies \theta=\dfrac{11\pi}{6}$
Thus, $(r, \theta, \phi)=(4, \dfrac{11\pi}{6},\dfrac{\pi}{6})$
