Why is it sufficient to set the polar angle between 0 and 180 degrees in a spherical coordinate system?

Question

In the spherical coordinate system

$x=r \sin \theta \cos \phi$

$y=r \sin \theta \sin \phi$

$z=r \cos \theta$

$\theta$ lies in $[0,\pi]$, while $\phi$ is in $[-\pi,\pi)$. Why do we not need for the sign of $\theta$?

Answer 1

We have

$$x=r \sin (-\theta) \cos \phi=r (-\sin \theta) \cos \phi=r \sin \theta (-\cos \phi)=r \sin \theta \cos(\phi+\pi)$$ $$y=r \sin (-\theta) \sin \phi=r (-\sin \theta) \sin \phi=r \sin \theta (-\sin \phi)=r \sin \theta \sin (\phi+\pi)$$ $$z=r \cos (-\theta)=r \cos \theta$$

So $(r,-\theta,\phi)$ represents the same point as $(r,\theta,(\phi+\pi )\pmod {2\pi})$

Answer 2

Imagine standing in the middle of your kitchen. You can turn around through a vertical axis(your phi) and with your arm by your side can raise or lower your hand(your theta). Then you can point to any object in your room without breaking your elbow (thats theta more than 180)

Also take care here, many sources flip theta and phi. I think in your definition theta is the angle to the z-axis (not really the polar angle). For most it is phi. Your definition is common but confusing. Much better is $z=r\cos(\phi)$ with $\phi$ ranging from 0 to 180. This is the source of the confusion.