Polar Coordinates

The Polar Coordinate System

A point is represented by $(r, \theta)$ where:

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

Converting Between Systems

Polar to Rectangular

$x = r \cos\theta$

$y = r \sin\theta$

Rectangular to Polar

$r = \sqrt{x^2 + y^2}$

$\tan\theta = \frac{y}{x}$

Note: When finding $\theta$, consider the quadrant:

• Quadrant I: $\theta = \arctan(y/x)$
• Quadrant II: $\theta = \arctan(y/x) + \pi$
• Quadrant III: $\theta = \arctan(y/x) + \pi$
• Quadrant IV: $\theta = \arctan(y/x) + 2\pi$ (or negative angle)

Multiple Representations

The same point can have different polar representations:

$(r, \theta) = (r, \theta + 2\pi n) \quad \text{for any integer } n$

$(r, \theta) = (-r, \theta + \pi)$

Common Polar Equations

Lines

Equation	Description
$\theta = \alpha$	Line through origin at angle $\alpha$
$r \cos\theta = a$	Vertical line $x = a$
$r \sin\theta = b$	Horizontal line $y = b$

Circles

Equation	Description
$r = a$	Circle centered at origin, radius $a$
$r = 2a \cos\theta$	Circle through origin, center $(a, 0)$
$r = 2a \sin\theta$	Circle through origin, center $(0, a)$

Rose Curves

$r = a \cos(n\theta)$

$r = a \sin(n\theta)$

• $n$ odd: $n$ petals
• $n$ even: $2n$ petals

Limaçons

$r = a + b \cos\theta$

$r = a + b \sin\theta$

Condition	Shape
$\frac{a}{b} < 1$	Inner loop
$\frac{a}{b} = 1$	Cardioid (heart)
$1 < \frac{a}{b} < 2$	Dimpled
$\frac{a}{b} \geq 2$	Convex

Cardioid

$r = a(1 + \cos\theta)$

$r = a(1 + \sin\theta)$

Lemniscate

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

$r^2 = a^2 \sin(2\theta)$

Figure-eight shape

Spiral of Archimedes

$r = a\theta$

Symmetry Tests

Symmetry about Polar Axis (x-axis)

Replace $\theta$ with $-\theta$. If equation unchanged, symmetric about polar axis.

Symmetry about $\theta = \frac{\pi}{2}$ (y-axis)

Replace $\theta$ with $\pi - \theta$. If equation unchanged, symmetric about y-axis.

Symmetry about Pole (origin)

Replace $r$ with $-r$, or $\theta$ with $\theta + \pi$. If equation unchanged, symmetric about origin.

Converting Equations

Rectangular to Polar

Use substitutions:

$x = r \cos\theta$

$y = r \sin\theta$

$x^2 + y^2 = r^2$

Example: $x^2 + y^2 = 9 \to r = 3$

Polar to Rectangular

Use substitutions:

$r^2 = x^2 + y^2$

$r \cos\theta = x$

$r \sin\theta = y$

$\tan\theta = \frac{y}{x}$

Example: $r = 4 \sin\theta$

Multiply by $r$: $r^2 = 4r \sin\theta$

Substitute: $x^2 + y^2 = 4y$

Complete square: $x^2 + (y-2)^2 = 4$

Circle with center $(0, 2)$ and radius $2$