Question

How do you convert the cartesian coordinate (−2, −9) into polar coordinates?

Answer 1

`(-2,-9)` [Cartesian] `=(sqrt(85),pi+arctan(9/2))` [Polar]

Explanation:

Polar coordinates are in the form `(r,theta)`
where `r` is the radial distance from the origin
and `theta` is the angle measured (in a counter clockwise direction) from the positive X-axis.

The radius can be calculated using the Pythagorean Theorem
`color(white)("XXXX")r = sqrt((-2)^2+(-9)^2) = sqrt(85)`

The angel `theta` is `pi` (half a rotation) plus and additional (reference) angle `hattheta`
where `tan(hattheta) = (-9)/(-2) = 9/2`
which implies
`color(white)("XXXX")hattheta=arctan(9/2)`
and
`color(white)("XXXX")theta=pi+arctan(9/2)`