Polar Coordinates and Graphs

By the end of this lesson, you will be able to convert between rectangular and polar coordinates, plot points in polar form, and graph basic polar curves (circles, roses, limaçons, lemniscates).

The Big Idea

The rectangular coordinate system (x, y) is great for horizontal and vertical relationships. The polar coordinate system $(r, \theta)$ is great for circular and rotational symmetry. Any point can be described in either system; many curves that are complex in rectangular form are beautifully simple in polar form.

Polar Coordinates

Every point is located by:

• $r$: distance from the pole (origin)
• $\theta$: angle from the polar axis (positive $x$-axis), measured counterclockwise

Non-uniqueness: $(r, \theta)$ and $(r, \theta + 2n\pi)$ are the same point. Also, $(-r, \theta)$ is the point at distance $r$ in the opposite direction.

Converting Between Systems

Polar → Rectangular:

$$x = r\cos\theta, \quad y = r\sin\theta$$

Rectangular → Polar:

$$r = \sqrt{x^2 + y^2}, \quad \theta = \tan^{-1}\left(\frac{y}{x}\right)$$

(Adjust $\theta$ for the quadrant.)

Example 1 — Conversion

Polar $(5, 2\pi/3)$ → Rectangular:

$x = 5\cos(2\pi/3) = 5(-1/2) = -5/2$

$y = 5\sin(2\pi/3) = 5(\sqrt{3}/2) = 5\sqrt{3}/2$

Rectangular: $(-5/2, 5\sqrt{3}/2)$

Rectangular $(-3, 3\sqrt{3})$ → Polar:

$r = \sqrt{9 + 27} = 6$

$\theta = \tan^{-1}(3\sqrt{3}/(-3)) = \tan^{-1}(-\sqrt{3})$; since $x < 0, y > 0$ → QII: $\theta = 2\pi/3$

Polar: $(6, 2\pi/3)$

Common Polar Curves

Curve	Polar equation	Shape
Circle (centered at pole)	$r = a$	Circle, radius $a$
Circle through pole	$r = 2a\cos\theta$ or $r = 2a\sin\theta$	Circle, diameter $2a$
Rose	$r = a\cos(n\theta)$ or $r = a\sin(n\theta)$	$n$ petals (odd $n$) or $2n$ petals (even $n$)