You need to show that if a place has (latitude, longitude) = $(p,q)$ then its coordinates are $$(R \cos p \cos q, R \cos p \sin q, R \sin p).$$ 
Steps in the
calculation
We need the distance $l$ on the surface between A and B.
First calculate the threedimensional coordinates of A and B
from the latitude and longitude of the two points.
Then calculate the distance $2d$ (imagine a tunnel straight
through the earth from A to B) using Pythagoras Theorem.
Use $R$, the radius of the Earth, and $d$ to find the angle
$\theta$ radians where $$\sin \theta = {d\over R}.$$ Calculate the
arc length $l$.
