Answer
The longitude and latitude for $\left( {\theta ,\varphi } \right) = \left( {\pi /8,7\pi /12} \right)$:
(15$^\circ $ S, 22.5$^\circ $ E).
The longitude and latitude for $\left( {\theta ,\varphi } \right) = \left( {4,2} \right)$:
(24.59$^\circ $ S, 130.82$^\circ $ W).
Work Step by Step
1. $\left( {\theta ,\varphi } \right) = \left( {\pi /8,7\pi /12} \right)$
We have $\left( {\theta ,\varphi } \right) = \left( {\pi /8,7\pi /12} \right) = \left( {22.5^\circ ,105^\circ } \right)$
The latitude of the point is $105^\circ - 90^\circ = 15^\circ $ south of the equator.
The longitude of the point is 22.5$^\circ $ to the east of Greenwich. Thus, the longitude and latitude for this point is (15$^\circ $ S, 22.5$^\circ $ E).
2. $\left( {\theta ,\varphi } \right) = \left( {4,2} \right)$
We have $\left( {\theta ,\varphi } \right) = \left( {4,2} \right) = \left( {229.18^\circ ,114.59^\circ } \right)$
The latitude of the point is $114.59^\circ - 90^\circ = 24.59^\circ $ south of the equator.
The longitude of the point is $360^\circ - 229.18^\circ = 130.82^\circ $ to the west of Greenwich. Thus, the longitude and latitude for this point is (24.59$^\circ $ S, 130.82$^\circ $ W).
