Answer
The spherical angles for Helsinki:
$\left( {\theta ,\phi } \right) = \left( {25.0^\circ ,29.9^\circ } \right)$.
The spherical angles for São Paulo:
$\left( {\theta ,\phi } \right) = \left( {313.48^\circ ,113.52^\circ } \right)$.
Work Step by Step
1. Helsinki, Finland (60.1$^\circ$ N, 25.0$^\circ$ E)
We have the latitude 60.1$^\circ$ N and the longitude 25.0$^\circ$ E.
Since the latitude of Helsinki is north of the equator, $60.1^\circ = 90^\circ - \phi $, So, $\phi = 29.9^\circ $.
Since the longitude of Helsinki lies to the east of Greenwich, $\theta = 25.0^\circ $.
Thus, the spherical angles $\left( {\theta ,\phi } \right)$ for Helsinki: $\left( {\theta ,\phi } \right) = \left( {25.0^\circ ,29.9^\circ } \right)$.
2. São Paulo, Brazil (23.52$^\circ$ S, 46.52$^\circ$ W)
We have the latitude 23.52$^\circ$ S and the longitude 46.52$^\circ$ W.
São Paulo's latitude is south of the equator, so $23.52^\circ = \phi - 90^\circ $ and $\phi = 113.52^\circ $.
Since 46.52$^\circ$ W refers to 46.52$^\circ$ in the negative $\theta$ direction, we have $\theta = 360^\circ - 46.52^\circ = 313.48^\circ $.
Thus, the spherical angles $\left( {\theta ,\phi } \right)$ for São Paulo: $\left( {\theta ,\phi } \right) = \left( {313.48^\circ ,113.52^\circ } \right)$.
