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PAGE 1

12.2 Polar Coordinates (continued)

Graphs of Polar Equations

Example: \( r = \cos(\theta) \)

#\( \theta \)\( r \)\( (r, \theta) \)
101\( (1, 0) \)
2\( \frac{\pi}{4} \)\( \frac{1}{\sqrt{2}} \approx 0.7 \)\( (\frac{1}{\sqrt{2}}, \frac{\pi}{4}) \)
3\( \frac{\pi}{2} \)0
4\( \frac{3\pi}{4} \)\( -\frac{1}{\sqrt{2}} \approx -0.7 \)
5\( \pi \)-1
6\( \frac{5\pi}{4} \)-0.7
7\( \frac{3\pi}{2} \)0
8\( \frac{7\pi}{4} \)0.7
9\( 2\pi \)1
Cartesian graph of r = cos(theta) showing a wave starting at (0,1) and crossing the theta-axis at pi/2 and 3pi/2.
The usual cosine graph we are used to seeing; this is the Cartesian graph.
PAGE 2

The Polar Graph

The Polar graph is more interesting.

Polar graph of r = cos(theta) forming a circle centered on the x-axis, passing through the origin and (1,0).
Circle (goes counterclockwise once)

Note: The graph includes dashed lines representing angles like \( \frac{\pi}{4} \). Points are labeled with numbers corresponding to the table from the previous page.

PAGE 3

Example: Polar Graphing

Equation: \( r = \sin(2\theta) \)

Point\( \theta \)\( r \)
100
2\( \frac{\pi}{8} \)\( \frac{1}{\sqrt{2}} \approx 0.7 \)
3\( \frac{\pi}{4} \)1
4\( \frac{3\pi}{8} \)\( \frac{1}{\sqrt{2}} \approx 0.7 \)
5\( \frac{\pi}{2} \)0
6\( \frac{5\pi}{8} \)\( -\frac{1}{\sqrt{2}} \approx -0.7 \)
7\( \frac{3\pi}{4} \)-1
8\( \frac{7\pi}{8} \)\( -\frac{1}{\sqrt{2}} \approx -0.7 \)
9\( \pi \)0
10\( \frac{9\pi}{8} \)\( \frac{1}{\sqrt{2}} \approx 0.7 \)
11\( \frac{10\pi}{8} \)...
\( 2\pi \)0

Note: The values repeat periodically.

A polar graph of the function r = sin(2 theta) plotted on x and y axes, resulting in a four-petaled rose curve.
Figure 1: Rose curve with 4 petals.
PAGE 4

General Properties of Rose Curves

\( r = \cos(n\theta) \) and \( r = \sin(n\theta) \) are roses (a circle is a one-petal rose).

  • If \( n \) is even \( \rightarrow 2n \) petals
  • If \( n \) is odd \( \rightarrow n \) petals
PAGE 5

Example: Polar Graphing

Equation: \[ r = 1 - 3 \cos(\theta) \]

Point\(\theta\)\(r\)
10-2
2\(\frac{\pi}{4}\)-1.12
3\(\frac{\pi}{2}\)1
4\(\frac{3\pi}{4}\)3.12
5\(\pi\)4
6\(\frac{5\pi}{4}\)3.12
7\(\frac{3\pi}{2}\)1
8\(\frac{7\pi}{4}\)-1.12
9\(2\pi\)-2
A polar graph of r = 1 - 3 cos(theta) forming a limaçon with an inner loop, plotted on Cartesian axes.
Figure 1: Limaçon with inner loop.

Note: This shape is called a limaçon (snail).

Transition from point 2 to 3: How? Insert additional point between 2 and 3.

PAGE 6

Example: Polar Graphing

Equation: \[ r = 1 - \sin(\theta) \]

Point\(\theta\)\(r\)
101
2\(\frac{\pi}{4}\)0.3
3\(\frac{\pi}{2}\)0
4\(\frac{3\pi}{4}\)0.3
5\(\pi\)1
6\(\frac{5\pi}{4}\)1.7
7\(\frac{3\pi}{2}\)2
8\(\frac{7\pi}{4}\)1.7
9\(2\pi\)1
A polar graph of r = 1 - sin(theta) forming a heart-shaped cardioid on Cartesian axes.
Figure 2: Cardioid (heart shape).

This shape is a cardioid (heart!).