What is the equivalent polar equation of $x^2 + (y-1)^2 = 1$?

Question

It's a question in the textbook that I have and I am having a hard time understanding it. How am I supposed to get the polar equation with this format?

Answer 1

Converting to polar, you have $x = r \cos(\theta)$ and $y = r \sin(\theta)$. So make these substitutions and simplify.

Answer 2

Just substitute: $x = r\cos(\theta)$ and $y = r\sin(\theta)$:

\begin{align*} r^2\cos^2(\theta) + (r\sin(\theta) - 1)^2 =& r^2\cos^2(\theta) + r^2\sin^2(\theta) - 2r\sin(\theta) + 1 = 1 \\ r^2 - 2r\sin(\theta) + 1 =& 1 \\ r^2 - 2r\sin(\theta) = 0 \\ r(r - 2\sin(\theta)) = 0 \\ r = 0, r = 2\sin(\theta) \end{align*}

Well, clearly, $r = 0$ is not correct (it's only mathematically correct because of the equations), so this leaves: $r = 2\sin(\theta)$

Answer 3

I would expand $(y-1)^2\to y^2-2y+1$ , making the equation $x^2+y^2-2y+1=1$ , which simplifies to $x^2+y^2=2y$ . $x^2+y^2$ can be replaced by $r^2$ , and $y$ can be replaced by $r\sin\theta$ , so $r^2=2r\sin\theta$ .