Translating Equations and Inequalities between Coordinate Systems Examples

Translate the equation x + y = 2x² into polar coordinates.

Replace x with r cos θ and y with r sin θ.

x + y = 2x²

r cos θ + r sin θ = 2 (r cos θ)²
r cos θ + r sin θ = 2r² cos² θ

One solution to this equation is r = 0. If r isn't zero, we can divide both sides of the equation by r to find

cos θ + sin θ = 2r cos² θ.

Since polar equations are usually written in the form

r = f(θ)

we rearrange this a bit more to find

.

Translate the inequality x² + y² > 1 into polar coordinates.

We can replace x² + y² with r² to find the polar inequality r > 1. This describes all points that lie outside the unit circle:

Translate the equation r cos θ + r sin θ = 2 into rectangular coordinates.

This requires a straightforward substitution.

If r cos θ + r sin θ = 2 then

x + y = 2.

Translate the inequality into rectangular coordinates, assuming .

The cos θ piece is difficult to translate, but if we multiply both sides of the inequality by cos θ, we find

r cos θ > 3

so x > 3.

The restriction on the values of θ ensures that cosθ isn't negative, this way we don't accidentally multiply the inequality by a negative number.

To go from polar to rectangular coordinates we need to rearrange the equation or inequality until we have terms that can be replaced with x, y, or x² + y².
We may need to multiply both sides of an equation or inequality by r or by cos θ . We may need to square both sides. We may need to do other things.