\begin{align*} x &= a \sin \theta \cos \varphi\\ y &= a \sin \theta \sin \varphi\\ z &= a \cos \theta \end{align*}
Given this is an equation of a sphere, how would I find its center and radius?
\begin{align*} x &= a \sin \theta \cos \varphi\\ y &= a \sin \theta \sin \varphi\\ z &= a \cos \theta \end{align*}
Given this is an equation of a sphere, how would I find its center and radius?
If you want to get an intuitive picture of what is going on with these equations, you might like to consider the maximum and minimum values of the $x, y, z$ coordinates.
Since the trig functions vary between $+1$, and $-1$, it is easy to see that the maximum and minimum values of $x, y, z$ are $a$ and $-a$, so the centre is halfway between the maximum and minimum i.e. at $(0,0,0)$ and then the radius is clear.
It is centered at $(x,y,z)=(0,0,0)$ and its radius $= a$
You can do this by inspection of the conversion between Cartesian and Spherical Coordinates:
$\begin{align*} x &= r \cos\phi \sin\theta\\ y &= r \sin\phi \sin\theta\\ z &= r\cos\theta \end{align*}$