Convert the equation into rectangular form

Question

The polar equation is $$r=2\sin(\theta)$$ so knowing what $r$ equals I can use that as one of the "$r$'s" in the equation $r^2=x^2+y^2$ $$2rsin(\theta)=x^2+y^2$$ $$2y=x^2+y^2$$ $$0=x^2+y^2-2y$$ This is where I am stuck because the final answer should be $x^2+(y-1)^2=1$ and I am both exhausted and dehydrated and can't figure it out.

Answer 1

Note, $$x^2+y^2-2y=x^2+(y^2-2y+1)-1=x^2+(y-1)^2-1=0$$

Answer 2

What you have done is correct, just add and subtract 1 to make it a perfect square.

$x^2 + y^2 - 2y =0 \implies x^2 + y^2 - 2y +1 = 1 \implies x^2 + (y-1)^2 = 1$ , which is same as the answer provided.