Question

How do you change (4, -1) from rectangular to cylindrical coordinates between [0, 2π)?

Answer 1

I will assume you meant "polar coordinates" since cylindrical coordinates are 3-dimensional and you have only supplied 2-dimensional Cartesian coordinates.

From the graph:
`r = sqrt(4^2 + (-1)^2) = sqrt(17)`
and
`theta = arctan((-1)/4)`

Note that standard arctan functions will return a value which will need to be adjusted to compensate for the point being in the IV quadrant so `theta` will fall in the `[0,2pi)` range.

Depending upon the version used you will typically get
`arctan((-1)/4) = -0.245` (radians)
or
`arctan((-1)/4) = -14.040` (degrees)

Add `2pi` or `360^o` respectively to get the "correct" answer.