Write the polar equation in rectangular form.
r = 6 sin θ

Y=-6x

Y=x-3

X^2+(y-3)^2=9

(X-3)^2+y^2=9

Step-by-step explanation:

The identities you need here are:

 and  

You also need to know that

x = rcosθ  and

y = rsinθ

to get this done.

We have

r = 6 sin θ

Let's first multiply both sides by r (you'll always begin these this way; you'll see why in a second):

r² = 6r sin θ

Now let's replace r² with what it's equal to:

x² + y² = 6r sin θ

Now let's replace r sin θ with what it's equal to:

x² + y² = 6y

That looks like the beginnings of a circle.  Let's get everything on one side because I have a feeling we will be completing the square on this:

Complete the square on the y-terms by taking half its linear term, squaring it and adding it to both sides.

The y linear term is 6.  Half of 6 is 3, and 3 squared is 9, so we add 9 in on both sides:

In the process of completing the square, we created within that set of parenthesis a perfect square binomial:

And there's your circle!  Third choice down is the one you want.

Fun, huh?