Question

How do you write the rectangular equation `x^2+(y-2)^2=4` in polar form?

Answer 1

Expand the square.
Use `r^2 = x^2+y^2` and `y = rsin(theta)` to convert.
Solve the quadratic to obtain an equation of `r` in terms of `theta`

Explanation:

Given: `x^2+(y-2)^2=4`

Here is a graph of the original equation:

Expand the square:

`x^2+y^2-4y+4=4`

Use `r^2 = x^2+y^2` and `y = rsin(theta)` to convert.

`r^2-4rsin(theta)+4=4`

Combine like terms:

`r^2-4rsin(theta)=0`

We can discard a common factor of r, because it is only the trivial root `r = 0`

`r-4sin(theta)=0`

`r = 4sin(theta)`

Here is a graph of the converted equation:

Please observe that the graphs are identical.