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=3sin⁡(θ)r = 3\sin(\theta)

θ\theta00π6\frac{\pi}{6}π3\frac{\pi}{3}π2\frac{\pi}{2}2π3\frac{2\pi}{3}5π6\frac{5\pi}{6}π\pi7π6\frac{7\pi}{6}4π3\frac{4\pi}{3}3π2\frac{3\pi}{2}5π3\frac{5\pi}{3}11π6\frac{11\pi}{6}2π2\pi
3sin⁡(θ)3\sin(\theta)001.51.52.62.6332.62.61.51.500−1.5-1.5−2.6-2.6−3-3−2.6-2.6−1.5-1.500


  • Continued example:

    • θ\theta00π6\frac{\pi}{6}π3\frac{\pi}{3}π2\frac{\pi}{2}2π3\frac{2\pi}{3}5π6\frac{5\pi}{6}π\pi
    3sin⁡(θ)3\sin(\theta)001.51.52.62.6332.62.61.51.500

    		θ\theta7π6\frac{7\pi}{6}4π3\frac{4\pi}{3}3π2\frac{3\pi}{2}5π3\frac{5\pi}{3}11π6\frac{11\pi}{6}2π2\pi
    3sin⁡(θ)3\sin(\theta)−1.5-1.5−2.6-2.6−3-3−2.6-2.6−1.5-1.500

    Symmetry of Polar Graphs

    • There are 3 ways that a polar graph can be symmetrical:
      • With respect to the line θ=π2\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,θ)(r, \theta) or (−r,−θ)(-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⁡θ\cos \theta, it is symmetrical with respect to the polar axis.
      • Method 3: If it is a function of sin⁡θ\sin \theta, it is symmetrical with respect to the line θ=π2\theta = \frac{\pi}{2}.

    Finding Points

    Maximum/Minimum Points

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

    Finding Zeros

    • To find the zeros of the function, let r=0r = 0 and find the values of θ\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=acos⁡θr = a \cos \theta or r=asin⁡θr = a \sin \theta
      • where a > 0 or a≤0a \leq 0
    Limaçons
    • r=a+bcos⁡θr = a + b \cos \theta or r=a+bsin⁡θr = a + b \sin \theta, where a and b are both positive
      • Limaçon with inner loop: a < b
      • Cardioid: a=ba = b
      • Dimpled limaçon: b < a < 2b
      • Convex limaçon: a≥2ba \geq 2b
    Roses
    • r=acos⁡(nθ)r = a \cos(n\theta) or r=asin⁡(nθ)r = a \sin(n\theta), where n≥2n \geq 2 is an integer
      • The rose has n petals if n is odd and 2n petals if n is even.
    Lemniscates
    • r2=a2cos⁡(2θ)r^2 = a^2 \cos(2\theta) or r2=a2sin⁡(2θ)r^2 = a^2 \sin(2\theta)
    Spirals of Archimedes
    • r=a+bθr = 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=acos⁡θr = a \cos \theta or r=asin⁡θr = a \sin \theta
      • Limaçons: r=a±bcos⁡θr = a \pm b \cos \theta or r=a±bsin⁡θr = a \pm b \sin \theta
      • Roses: r=acos⁡(nθ)r = a \cos(n\theta) or r=asin⁡(nθ)r = a \sin(n\theta)
        • n is odd à n petals
        • n is even à 2n petals
      • Lemniscates: r2=a2cos⁡(2θ)r^2 = a^2 \cos(2\theta) or r2=a2sin⁡(2θ)r^2 = a^2 \sin(2\theta)
      • Spirals of Archimedes: r=a+bθr = 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=6sin⁡4θr = 6 \sin 4\theta

    Try These… Match each equation with its graph.

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