I know its a little silly, but I got the wrong sign several times. Just to be clear, $z=r\cos(\phi), -\frac{\pi}{2}\leq\phi\leq\frac{\pi}{2}$ when converting from cartesian to spherical. So, how do I determine the sign? Thanks!
2026-03-30 16:55:44.1774889744
Finding the sign of $\phi$ in spherical coordinates
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The spherical coordinate system I learned has $z=r \cos \theta$ with $0 \le \theta \le \pi$. For that range of $\phi$ you need to use $z=r \sin(\phi)$, in which case you can get the sign. I have seen geographers use this, though they usually use $\lambda$ for latitude