Answer
a) $(2, \dfrac{3\pi}{2},\dfrac{\pi}{2})$
b) $(2, \dfrac{3\pi}{4},\dfrac{3\pi}{4})$
Work Step by Step
Convert rectangular coordinates to spherical coordinates.
$x=\rho \sin \phi \cos \theta; y=\rho \sin \phi \sin \theta;z=\rho \cos \phi$
and $\rho=\sqrt {x^2+y^2+z^2}$
a) Here, $\rho=2$
$\cos \phi =\dfrac{0}{2} $
or, $\phi=\dfrac{\pi}{2}$
and $\cos \theta=\dfrac{0}{2 \sin \dfrac{\pi}{2}} $
or, $ \theta=\dfrac{3\pi}{2}$
Thus, $(2, \dfrac{3\pi}{2},\dfrac{\pi}{2})$
b) Convert rectangular coordinates to spherical coordinates.
$x=\rho \sin \phi \cos \theta; y=\rho \sin \phi \sin \theta;z=\rho \cos \phi$
and $\rho=\sqrt {x^2+y^2+z^2}$
Here, $\rho=2$
$\cos \phi =\dfrac{-\sqrt 2}{2}$
or, $\phi=(\dfrac{3 \pi}{4}) $; and $\cos \theta=\dfrac{-1}{2 \sin (\dfrac{3\pi}{4})}$
or, $ \theta=\dfrac{3\pi}{4}$
so, we have $(2, \dfrac{3\pi}{4},\dfrac{3\pi}{4})$
