# Finding the sign of $\phi$ in spherical coordinates

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!

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If $\phi$ here is supposed to be co-latitude, the version of spherical coordinates that I am accustomed to has it go from $0$ (North Pole) to $\pi$ (South Pole). It seems as if you're measuring from the equator instead of from the poles? – J. M. Feb 15 '12 at 14:33
Yes, exactly. But the method would be the same with little changes, wouldn't it? – yotamoo Feb 15 '12 at 14:42
If, as you claim, the changes are "little", then you can figure out how to correct $\phi=\arccos\frac{z}{r}$, which assumes measurement from the poles, to your preferred convention, no? – J. M. Feb 15 '12 at 14:46
I just happen to be a TA in a calculus course this semester and we just got to spherical coordinates. We teach that $0\leq\phi\leq\pi$ and $z=r\cos\phi$. – Asaf Karagila Jun 17 '12 at 6:37

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