Question

How do you convert `x^2 + y^2 = 4y` to polar form?

Answer 1

The polar equation is `r=4sintheta`

Explanation:

Apply the following to convert from rectangular coordinates `(x,y)` to polar coordinates `(r,theta)` :

`{(x=rcostheta),(y=rsintheta),(x^2+y^2=r^2), (theta=arctan(y/x)):}`

Here, we have

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

`r^2=4rsintheta`

As `r!=0`

`r=4sintheta`

Answer 2

`r=4sintheta`

Explanation:

The conversion from Rectangular to Polar:
`x=rcostheta`
`y=rsintheta`

Substitute for `x` and `y`:
`(rcostheta)^2+(rsintheta)^2= 4(rsintheta)`

`r^2cos^2theta+r^2sin^2theta= 4rsintheta`

Factor out the `r^2`:
`r^2(cos^2theta+sin^2theta)= 4rsintheta`

Apply pythagorean identity:
`sin^2theta+cos^2theta=1`
`r^2(1)= 4rsintheta`

Set the expression equal to 0:
`r^2-4rsintheta=0`

Factor out the `r`:
`r(r-4sintheta)=0`

At this point either `r=0` or `r-4sintheta=0`, let's solve the second to get a meaningful answer:

`r=4sintheta`