Answer
$x^2 + (y-3)^2 = 9$
See graph below.
Work Step by Step
The question asks for a graph of the polar equation and to convert the equation into rectangular coordinates
Given $r = 6 \sin θ$
Table of Values:
$(0, 0)$
$(\frac{\pi}{4}, 3 \sqrt 2)$
$(\frac{\pi}{2}, 6)$
$(\frac{3\pi}{4}, 3 \sqrt 2)$
$(\pi, 0)$
$(\frac{5\pi}{4}, -3 \sqrt 2)$
$(\frac{3\pi}{2}, -6)$
$(\frac{7\pi}{4}, -3 \sqrt 2)$
See graph below.
To convert the equation into rectangular coordinates, the following equation can be used:
$r^2 = x^2 + y^2$ $y = r \sin θ$
So $r = 6 \sin θ$
$r^2 = 6r \sin θ$
$x^2 + y^2 = 6y$
$x^2 + y^2 - 6y = 0$
$x^2 + y^2 - 6y + 9= 0 + 9$
$x^2 + (y-3)^2 = 9$
Circle centered at (0,3) with radius 3
