Education Technology

Exploring Polar Equations

Published on 06/09/2008

Activity Overview

In this activity, students will explore the polar equations
r(θ) = a + bcos(nθ) and r(θ) = a + bsin(nθ). They learn that by changing values a, b, and n, different polar curves can be generated.

Before the Activity

  • See the attached PDF file for detailed instructions for this activity
  • Print pages 89 - 95 from the attached PDF file for your class
  • During the Activity

    Distribute the pages to the class.

    Follow the Activity procedures:

  • Define r(θ) = bcos(θ) for b = 2 and graph it
  • Change value of b to 3, -4, and -5 and observe the different graphs
  • Observe that change in the absolute value of b results in an increase in the radius of the circle; for negative value of b the circle is reflected about the y-axis


  • Rose Polar Curve
  • Define r(θ) = 4cos(nθ) for n = 1 and graph it
  • Change the value of n to 2, 3, and 4 and observe the different graphs
  • Note that if n is odd, the number of rose leaves is equal to n; if n is even, the number of leaves is twice n
  • Observe that as the value of b increases the length of each leaf increases; for negative value of b the rose is reflected about the y-axis


  • Limacon Curve
  • Define r(θ) = a + bcos (θ) for a = 1, b = 2, n = 1 and graph it
  • Change the values of a and b and observe the graphs
  • Observe that the absolute value of b is always greater than the absolute value of a, the greater the sum of the absolute values of a and b the larger the outer loop
  • Also, for negative value of b the limacon is reflected about the y-axis


  • Cardioid Curve
  • Define r(θ) = a + bcos(θ) for a = 2, b = 2, n = 1 and graph it
  • For a = b = 4, and a = b = 5 graph the function
  • Observe that the absolute values of a and b are equal and their sum results in a larger loop
  • Change the value of b to -3 and graph the function
  • Observe that for negative value of b the cardioid is reflected about the y-axis
  • Solve other polar coordinate problems
  • After the Activity

    Students complete the Student Activity sheet.

    Review student results:

  • As a class, discuss questions that appeared to be more challenging
  • Re-teach concepts as necessary