The Earth is centered at the origin of a 3d graph, such that the xy-plane has the equator, the xz-plane has the prime meridian, and the north pole is on the positive z-axis.
Find the Cartesian coordinates of the town Beta which is located 40 degrees north of the equator and 15 degrees east of the prime meridian.
I am so confused by this problem. I tried to parameterize this and it led me nowhere.
I express my answer in terms of the formalism of spherical coordinates illustrated there https://en.wikipedia.org/wiki/File:3D_Spherical.svg.
It's all about to recognise $(r,\theta,\phi)$ identify town Beta and converting them into $(x,y,z)$. To do that imagine to have a compass and looking Earth like in this picture https://en.wikipedia.org/wiki/File:Spherical_coordinate_system.svg .
Then saying that Beta is located $40$° north of the equator means that the complementary angle of $\theta$ is $40$°, then $\theta=90°-40°=50°$. Now, moving $\phi$ to east from the first meridian means that it moves in its positive direction, i.e. in your right direction if you watch it as in the picture below. Then, $(r,\theta,\phi)=(R_E,50°,15°)$, where $R_E:=$ Earth radius. So using the conversion relations, cartesian coordinates are: $(x,y,z)=\bigl( R_E \sin \theta \cos \phi, R_E \sin \theta \sin \phi, R_E \cos \theta \bigr)$, where values found below have to be use.