Question

How do you convert `(-2, -2sqrt{3})`into polar coordinates?

Answer 1

`(4, (4pi)/3)` (radians) or `(4, 240^@)` (degrees)

Explanation:

Rectangular `->` Polar: `(x, y) -> (r, theta)`

• Find `r` (radius) using `r = sqrt(x^2 + y^2)`
• Find `theta` by finding the reference angle: `tantheta = y/x` and use this to find the angle in the correct quadrant

`r = sqrt((-2)^2 + (-2sqrt3)^2)`

`r = sqrt(4+(4*3))`

`r = sqrt(4+12)`

`r = sqrt(16)`

`r = 4`

Now we find the value of `theta` using `tantheta = y/x`.

`tantheta = (-2sqrt3)/-2`

`tantheta = sqrt3`

`theta = tan^-1(sqrt3)`

`theta = (pi)/3` or `(4pi)/3`

To determine which one it is, we have to look at our coordinate `(-2, -2sqrt3)`. First, let's graph it:

As you can see, it is in the third quadrant. Our `theta` has to match that quadrant, meaning that `theta = (4pi)/3`.

From `r` and `theta`, we can write our polar coordinate:
`(4, (4pi)/3)` (radians) or `(4, 240^@)` (degrees)

Hope this helps!