Polar Coordinates and Complex Numbers Notes

Graphs of Polar Equations

Chapter 8 - Polar Coordinates and Complex Numbers

Learning Objectives

• Graph polar equations
• Identify and graph classical curves

Starter Activity

• Polar coordinate system
• Polar coordinates
• Polar axis
• Pole
• The origin of a polar coordinate system
• A point represented with a distance and an angle
• A grid of concentric circles
• A horizontal ray directed right from the pole

Plotting Points

• Like a Cartesian coordinate system, we can find the graph of a polar function by creating a table of values and plotting points.




• Page 498, question 4




• Example:
  r = 3\sin(\theta)

• There are 3 ways that a polar graph can be symmetrical:
  • With respect to the line \theta = \frac{\pi}{2}
  • With respect to the polar axis
  • With respect to the pole

Tests for Symmetry

• Checking for symmetry: We can check if a polar equation will give us a symmetrical graph.
  • Method 1: If (r, \theta) or (-r, -\theta) gives us an equivalent equation, the graph will be symmetric with respect to the polar axis.
  • Method 2: If it is a function of \cos \theta, it is symmetrical with respect to the polar axis.
  • Method 3: If it is a function of \sin \theta, it is symmetrical with respect to the line \theta = \frac{\pi}{2} .

Finding Points

Maximum/Minimum Points

• Finding maximum values depends on when the distance between the point (r, \theta) and the pole is the greatest.
• Page 495, question 4A
• Maximum value usually happens when \sin \theta or \cos \theta is 1.
• Minimum value usually happens when \sin \theta or \cos \theta is -1.

Finding Zeros

• To find the zeros of the function, let r = 0 and find the values of \theta.
• Page 495, question 4A

Classic Polar Curves

• There are 5 special types of polar curves that you need to know:
  • Circles
  • Limaçons
  • Roses
  • Lemniscates
  • Spirals of Archimedes

Circles

• r = a \cos \theta or r = a \sin \theta
  • where a > 0 or a \leq 0

Limaçons

• r = a + b \cos \theta or r = a + b \sin \theta, where a and b are both positive
  • Limaçon with inner loop: a < b
  • Cardioid: a = b
  • Dimpled limaçon: b < a < 2b
  • Convex limaçon: a \geq 2b

Roses

• r = a \cos(n\theta) or r = a \sin(n\theta), where n \geq 2 is an integer
  • The rose has n petals if n is odd and 2n petals if n is even.

Lemniscates

• r^2 = a^2 \cos(2\theta) or r^2 = a^2 \sin(2\theta)

Spirals of Archimedes

• r = a + b\theta
  • \theta > 0, r = a + b
  • \theta < 0, r = a + b

Special Types Summary

• There are 5 special types of polar curves that you need to know:
  • Circles: r = a \cos \theta or r = a \sin \theta
  • Limaçons: r = a \pm b \cos \theta or r = a \pm b \sin \theta
  • Roses: r = a \cos(n\theta) or r = a \sin(n\theta)
    • n is odd à n petals
    • n is even à 2n petals
  • Lemniscates: r^2 = a^2 \cos(2\theta) or r^2 = a^2 \sin(2\theta)
  • Spirals of Archimedes: r = a + b\theta

Identifying Curves

• By looking at the given rules, we should be able to write polar equations when given a graph of a polar function.
• Page 498, questions 37

Starter Activity - Select the correct graph for r = 6 \sin 4\theta

Try These… Match each equation with its graph.

• 50. r = 1 - 4 cos 3θ
• 51. r = 1 - 4 sin 4θ
• 52. r = 1 + 3 sin 3θ
• 53. r = 1 + 3 cos 4θ
• Answers:
  • 50) c
  • 51) a
  • 52) d
  • 53) b