Question

How do you convert `(-4,-4)` into polar form?

Answer 1

`(4sqrt2, 225°)`

Explanation:

To convert between polar and rectangular, these formulas will help you out. (Link to the webpage I got the image from is in gray.)

So we can see that where the Cartesian/rectangular ordered pairs follow the `(x, y)` format, polar ordered pairs follow `(r, theta)`.

The steps here can vary, depending on whether you want to find `theta` or `r` first, but we'll just find `theta` first for this answer.

`tantheta = (-4)/-4 = 1`
`arctan(1) = 45°`

We should note, however, that since `x` and `y` are both negative, our point is in quadrant `III`. `45°` is just the reference angle. `theta` is actually `225°`.

We can use one of the left-hand formulas to find `r` now.

`-4 = rcos225°`

`-4 = r((-sqrt2)/2)`

`r = -4 * 2/(-sqrt2) = (-8)/(-sqrt2) = (-sqrt2^6)/(-sqrt2) = sqrt2^5 = 4sqrt2`

So we have found our answer: `(-4, -4)` in polar form is `(4sqrt2, 225°)`.

Of course, the way you write your answer is going to depend on whether they want `theta` in radians or degrees and whether they want you to be exact with `r` or round, but you get the gist, hopefully.