Question

How do you convert `(0, −4)` to polar form?

Answer 1

`(4, -pi/2)`

Explanation:

To convert from Cartesian Coordinates `(x,y)` to Polar Coordinates `(r,theta`):
Where `r = sqrt( x^2 + y^2 )` and `theta = arctan ( y / x )`

In this example: `x=0 and y=-4`

`:. r = sqrt(0^2+(-4)^2) =4`

`theta = arctan (-4/0)` which is undefined

Consider:
`theta = arctan(lim_"x->0" -4/x)`

`= lim_"x->-oo" arctan (x)`

`= -pi/2` [* See graph of `arctan x` below]

Hence, `(0, -4)` in polar coordinates is: `(4, -pi/2)`

*Graph of `arctan x`

graph{arctan x [-16.01, 16.03, -8.01, 8]}