Answer
$r=8\sin\theta$ (see graph)
Work Step by Step
We are given the circle:
Center: $(0,4)$
Radius: $4$
The rectangular equation of the circle is:
$(x-0)^2+(y-4)^2=4^2$
$x^2+(y-4)^2=16$
Write the equation in polar form using the formulas:
$x=r\cos\theta$
$y=r\sin\theta$
$x^2+y^2=r^2$
$(r\cos\theta)^2+(r\sin\theta-4)^2=16$
$r^2\cos^2\theta+r^2\sin^2\theta-8r\sin\theta+16=16$
$r^2(\cos^2\theta+\sin^2\theta)-8r\sin\theta=0$
$r^2-8r\sin\theta=0$
$r(r-8\sin\theta)=0$
$r-8\sin\theta=0$
$r=8\sin\theta$