We have a sphere with $r=1$, and we want the coordinates of $C$. $A$ is the north pole, and $AB$ is our prime meridian. See picture:
I'm familiar with an $(x,y,z)$ coordinate system, but not so much with polar coordinates. My booklet says that we can find the coordinates of $C$ using:
$$ x = \sin(b) \cdot \cos(A)$$ $$y = \sin(b) \cdot \sin(A)$$ $$z = \cos(b)$$
But it doesn't explain how, and I can't seem to figure it out myself (as this is my introduction to spherical geometry/trigonometry). Can anybody explain how this makes sense?
LettThe arc length be $l$ and so $$l=r\theta=\theta \implies b=\angle AOC$$ Then $$z=OC\cdot \cos \angle AOC=1\cos b$$ and $$x=OC\cdot \sin \angle AOC \cos \angle COB=1\sin b \cos a$$