In the polar coordinate system, each point is determined by:

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

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

Conventions

• $r$ can be positive or negative
  • If $r > 0$, move $r$ units in the direction of $\theta$
  • If $r < 0$, move $|r|$ units in the opposite direction of $\theta$
• $\theta$ is typically measured in radians
• Multiple representations: $(r, \theta)$, $(r, \theta + 2\pi n)$, $(-r, \theta + \pi)$ all represent the same point

Conversion Formulas

Polar to Rectangular

Given $(r, \theta)$ in polar: $x = r\cos(\theta)$ $y = r\sin(\theta)$

Rectangular to Polar

Given $(x, y)$ in rectangular: $r = \sqrt{x^2 + y^2}$ $\theta = \arctan(\frac{y}{x})$ (adjust for quadrant)

Or use: $\tan(\theta) = \frac{y}{x}$

Important: When finding $\theta$:

• Use $\arctan(\frac{y}{x})$ as starting point
• Adjust based on the quadrant of $(x, y)$
• Quadrant I: $\theta = \arctan(\frac{y}{x})$
• Quadrant II: $\theta = \pi + \arctan(\frac{y}{x})$
• Quadrant III: $\theta = \pi + \arctan(\frac{y}{x})$
• Quadrant IV: $\theta = 2\pi + \arctan(\frac{y}{x})$ or $\theta = \arctan(\frac{y}{x})$ (negative)

Common Polar Graphs

Circles

• $r = a$: Circle centered at origin with radius $a$
• $r = 2a\cos(\theta)$: Circle with diameter $2a$ on the polar axis
• $r = 2a\sin(\theta)$: Circle with diameter $2a$ perpendicular to polar axis

Lines

• $\theta = \alpha$: Line through origin at angle $\alpha$
• $r\cos(\theta) = a$: Vertical line $x = a$
• $r\sin(\theta) = a$: Horizontal line $y = a$

Limaçons

General form: $r = a \pm b\cos(\theta)$ or $r = a \pm b\sin(\theta)$

• If $\frac{a}{b} < 1$: Inner loop
• If $\frac{a}{b} = 1$: Cardioid (heart-shaped)
• If $1 < \frac{a}{b} < 2$: Dimpled limaçon
• If $\frac{a}{b} \geq 2$: Convex limaçon

Rose Curves

General form: $r = a\cos(n\theta)$ or $r = a\sin(n\theta)$

• If $n$ is odd: $n$ petals
• If $n$ is even: $2n$ petals
• Length of each petal: $|a|$

Lemniscates

• $r^2 = a^2\cos(2\theta)$: Figure-eight along polar axis
• $r^2 = a^2\sin(2\theta)$: Figure-eight at $45°$

Spirals

• $r = a\theta$: Archimedean spiral
• $r = ae^{b\theta}$: Exponential spiral

Symmetry in Polar Graphs

Test for symmetry to help sketch graphs:

1. Symmetry about the polar axis (x-axis):

   • Replace $(r, \theta)$ with $(r, -\theta)$
   • If equation unchanged, symmetric about polar axis

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

   • Replace $(r, \theta)$ with $(r, \pi - \theta)$ or $(-r, -\theta)$

3. Symmetry about the pole (origin):

   • Replace $(r, \theta)$ with $(-r, \theta)$ or $(r, \theta + \pi)$

Graphing Strategy

1. Identify the type of polar curve
2. Check for symmetry
3. Create a table of values for $\theta$ from $0$ to $2\pi$
4. Plot key points and note special values
5. Sketch the curve connecting points smoothly
6. Consider domain restrictions if $r^2$ appears (must have $r^2 \geq 0$)

Use conversion formulas: $x = r\cos(\theta)$ $y = r\sin(\theta)$

Find $x$: $x = 3\cos(\frac{2\pi}{3}) = 3 \cdot (-\frac{1}{2}) = -\frac{3}{2}$

Find $y$: $y = 3\sin(\frac{2\pi}{3}) = 3 \cdot \frac{\sqrt{3}}{2} = \frac{3\sqrt{3}}{2}$

Answer: $(-\frac{3}{2}, \frac{3\sqrt{3}}{2})$

Verification:

• Check distance: $r = \sqrt{(-\frac{3}{2})^2 + (\frac{3\sqrt{3}}{2})^2} = \sqrt{\frac{9}{4} + \frac{27}{4}} = \sqrt{9} = 3$ ✓
• Check angle: Point is in Quadrant II with correct ratio ✓

Step 1: Identify the curve type

This is a limaçon of the form $r = a + b\cos(\theta)$ where $a = 2$ and $b = 2$.

Since $\frac{a}{b} = \frac{2}{2} = 1$, this is a cardioid (heart-shaped).

Step 2: Check for symmetry

Test symmetry about the polar axis (x-axis): Replace $\theta$ with $-\theta$: $r = 2 + 2\cos(-\theta) = 2 + 2\cos(\theta)$

Equation unchanged, so symmetric about the polar axis ✓

Step 3: Create table of values

Step 4: Key features

• Maximum $r$: $r = 4$ when $\theta = 0$ (rightmost point)
• Minimum $r$: $r = 0$ when $\theta = \pi$ (cusp at origin)
• Axis of symmetry: Polar axis (x-axis)
• Shape: Heart-shaped curve pointing right
• Key points in rectangular coordinates:
  • $\theta = 0$: $(4, 0)$
  • $\theta = \frac{\pi}{2}$:

Sketch description: The curve starts at $(4, 0)$, curves upward and left, passes through $(0, 2)$, continues to the origin at $\theta = \pi$ forming a cusp, then curves downward through $(0, -2)$, and returns to $(4, 0)$. The overall shape resembles a heart lying on its side.