Question

How do I convert Cartesian coordinates to polar coordinates?

Answer 1

Let's look at the trig formulas SYR, CXR, TYX:

`sin theta = y/r`
`cos theta = x/r`
`tan theta = y/x`

Since we are given the Cartesian coordinates, we are given `x` and `y`. For polar coordinates, we need to figure out `r` and `theta`. `r` is easy, we just use Pythagorean:

`r=sqrt(x^2+y^2)`

To figure out `theta`, I like to use cosine because the range of arccosine is in quadrants I and II and adjusting `theta'` is easier. So,

`theta'=cos^(-1)x/r`

If `y>=0` then `theta=theta'`.
If `y<0` then `theta=2 pi - theta'` (in radians) or `theta=360-theta'` (in degrees).

Our final answer is `(r, theta)`.

Let's look at a concrete example: Convert `(-3, 3sqrt3)` to polar coordinates:

`r=sqrt((-3)^2+(3sqrt3)^2)=sqrt(36)=6`
`theta'=cos^(-1)((-3)/6)=(2pi)/3`
`y<0` so, `theta=2pi-(2pi)/3=(4pi)/3`

So the polar coordinates are `(6, (4pi)/3)`.