The Coordinate Plane and the Polar Plane

A coordinate system is just an agreed way of saying where a point is. Every system needs a reference point and a fixed rule. The Cartesian plane uses two distances; the polar plane uses one distance and one angle. Same point, two descriptions.

Concept Two ways to locate the same point

Cartesian:

(x,\,y)

— how far across, how far up.

Polar:

(r,\,\theta)

— how far from the pole, and in which direction.

The point drawn here is

(3,\,4)

in Cartesian form and

(5,\,53.1°)

in polar form.

Theorem The conversion formulas

Cartesian to polar:

r = \sqrt{x^2 + y^2}, \qquad \theta = \tan^{-1}\!\frac{y}{x}

Polar to Cartesian:

x = r\cos\theta, \qquad y = r\sin\theta

Example Converting the point (3, 4)

r = \sqrt{3^2 + 4^2} = \sqrt{25} = 5

\theta = \tan^{-1}\!\dfrac{4}{3} \approx 53.1°

⟹ (3, 4) becomes (5, 53.1°)

Going back:

x = 5\cos 53.1° \approx 3

and

y = 5\sin 53.1° \approx 4

⟹ the two forms describe the same point

Example Equations of a circle

A circle of radius 3 centred at the origin:

Cartesian:

x^2 + y^2 = 9

Polar:

r = 3

⟹ every point is at distance 3 — the polar form needs no algebra at all

Now the other direction — turn the polar equation $r = \cos\theta$ into Cartesian form. Multiply both sides by $r$ :

r^2 = r\cos\theta

x^2 + y^2 = x

x^2 - x + y^2 = 0

, and completing the square gives

\left(x - \tfrac{1}{2}\right)^2 + y^2 = \tfrac{1}{4}

⟹ a circle of radius ½ centred at (½, 0)

Reference Which system to reach for

Situation	Better choice
Straight lines, rectangles, simple algebra	Cartesian
A circle whose centre is not the origin	Cartesian
Rotation, angles, circles centred at the origin	Polar
Spirals, waves and vibrations	Polar

Summary

Cartesian coordinates (x, y) give horizontal and vertical distances; polar coordinates (r, θ) give a distance and a direction.
Cartesian to polar: r = √(x² + y²) and θ = tan⁻¹(y ÷ x).
Polar to Cartesian: x = r cos θ and y = r sin θ.
Choose the system that makes the shape simple — r = 3 beats x² + y² = 9 for a circle about the origin.