Question

What is the polar form of `(-200,10)`?

Answer 1

`(10sqrt401, 3.09)`

Explanation:

To convert this rectangular coordinate `(x,y)` to a polar coordinate `(r, theta)`, use the following formulas:

`r^2=x^2+y^2`
`tan theta=y/x`

`r^2=(-200)^2+(10)^2`
`r^2=40100`
`r=sqrt40100`
`r=10sqrt401`

`tan theta=y/x`

`tan theta=10/-200`
`theta=tan^-1(10/-200)`
`theta~~-0.05`

The angle `-0.05` radians is in Quadrant `IV`, while the coordinate `(-200, 10)` is in Quadrant `II`. The angle is wrong because we used the `arctan` function, which only has a range of `[-pi/2, pi/2]`. To find the correct angle, add `pi` to `theta`.

`-0.05 + pi = 3.09`

So, the polar coordinate is `(10sqrt401, 3.09)` or `(200.25, 3.09)`.