Polar Coordinates

The Polar Coordinate System

In the polar system, a point is located by:

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

A point is written as $(r, \theta)$.

Conversion Between Systems

Polar to Rectangular

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

Rectangular to Polar

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

Important: Adjust $\theta$ based on the quadrant of $(x, y)$.

Common Polar Graphs

Circle

$r = a \quad \text{(centered at origin, radius } a\text{)}$ $r = a\cos\theta \quad \text{(centered at } (a/2, 0)\text{, radius } a/2\text{)}$ $r = a\sin\theta \quad \text{(centered at } (0, a/2)\text{, radius } a/2\text{)}$

Line

$\theta = c \quad \text{(line through origin at angle } c\text{)}$

Classic Polar Curves

Rose Curves: $r = a\cos(n\theta)$ or $r = a\sin(n\theta)$

• $n$ odd → $n$ petals
• $n$ even → $2n$ petals

Cardioids: $r = a(1 + \cos\theta)$ or $r = a(1 + \sin\theta)$

• Heart-shaped curve that passes through the origin

Limaçons: $r = a + b\cos\theta$

• $|b| > |a|$: inner loop
• $|b| = |a|$: cardioid
• $|a| > |b|$: dimpled or convex

Lemniscate: $r^2 = a^2\cos(2\theta)$

• Figure-eight (infinity symbol) shape

Spiral of Archimedes: $r = a\theta$

• Spirals outward as $\theta$ increases

Converting Equations

Rectangular → Polar

Use $x = r\cos\theta$, $y = r\sin\theta$, and $x^2 + y^2 = r^2$.

Example: $x^2 + y^2 = 4x$ → $r^2 = 4r\cos\theta$ → $r = 4\cos\theta$

Polar → Rectangular

Multiply by $r$ or use identities to introduce $x$, $y$, $r^2$.

Example: $r = 2\sin\theta$ → $r^2 = 2r\sin\theta$ → $x^2 + y^2 = 2y$ → $x^2 + (y-1)^2 = 1$

Symmetry Tests

• Polar axis ($x$-axis): Replace $\theta$ with $-\theta$
• Line $\theta = \pi/2$ ($y$-axis): Replace $\theta$ with $\pi - \theta$
• Pole (origin): Replace $r$ with $-r$

Complex Numbers in Polar Form

$z = r(\cos\theta + i\sin\theta) = re^{i\theta}$

De Moivre's Theorem

$z^n = r^n(\cos(n\theta) + i\sin(n\theta))$

AP Precalculus Tip: Polar coordinates are heavily tested in Unit 4. Know conversions, curve types, and how to identify graphs from equations.