Question

How do you convert `r = 2 sin theta` into cartesian form?

Answer 1

Make use of a few formulas and do some simplification. See below.

Explanation:

When dealing with transformations between polar and Cartesian coordinates, always remember these formulas:

• `x=rcostheta`
• `y=rsintheta`
• `r^2=x^2+y^2`

From `y=rsintheta`, we can see that dividing both sides by `r` gives us `y/r=sintheta`. We can therefore replace `sintheta` in `r=2sintheta` with `y/r`:
`r=2sintheta`
`->r=2(y/r)`
`->r^2=2y`

We can also replace `r^2` with `x^2+y^2`, because `r^2=x^2+y^2`:
`r^2=2y`
`->x^2+y^2=2y`

We could leave it at that, but if you're interested...

Further Simplification
If we subtract `2y` from both sides we end up with this:
`x^2+y^2-2y=0`

Note that we can complete the square on `y^2-2y`:
`x^2+(y^2-2y)=0`
`->x^2+(y^2-2y+1)=0+1`
`->x^2+(y-1)^2=1`

And how about that! We end up with the equation of a circle with center `(h,k)->(0,1)` and radius `1`. We know that polar equations of the form `y=asintheta` form circles, and we just confirmed it using Cartesian coordinates.