Question

How do you convert `y = -x` to polar form?

Answer 1

Half-line `theta=(3pi)/4`, pole r=0 and the opposite half-line `theta=(7pi)/4`.

Explanation:

Cartesian `( x, y ) = ( r cos theta, r sin theta )` is `( r, theta )`, in polar form.

So, the equation `y = - x` becomes

`r sin theta=r cos theta`.

The solutions are `r = 0 and sin theta = - cos theta`

Further, in [0, `2pi`], `theta=(3pi)/4 and theta=(7pi)/4` make `sin theta = - cos theta`.

Importantly, r = 0 should be included, as .`theta=(3pi)/4 and theta=(7pi)/4` represent radial lines, sans r= 0.

In polar coordinates, r = 0 represents a null vector that has the distinction of associating with any direction..In the absence of r = 0, the line has discontinuity at the pole.

So, unless necessitated, polar form of the equation of a straight line is to be avoided. .