Polar Coordinates

Abstract

This lesson is designed to introduce students to graphing functions in polar coordinates.

Objectives

Upon completion of this lesson, students will:

  • Have a basic understanding of the graphs of trigonometric functions in polar coordinates
  • Understand how certain changes to the function affect the graph

Standards Addressed

Grade 9

  • Functions and Relationships

    • The student demonstrates conceptual understanding of functions, patterns, or sequences including those represented in real-world situations.
    • The student demonstrates algebraic thinking.

Grade 10

  • Functions and Relationships

    • The student demonstrates conceptual understanding of functions, patterns, or sequences including those represented in real-world situations.
    • The student demonstrates algebraic thinking.

Grades 8-12

  • Mathematical Analysis

    • 1.0 Students are familiar with, and can apply, polar coordinates and vectors in the plane. In particular, they can translate between polar and rectangular coordinates and can interpret polar coordinates and vectors graphically.

  • Trigonometry

    • 15.0 Students are familiar with polar coordinates. In particular, they can determine polar coordinates of a point given in rectangular coordinates and vice versa.
    • 16.0 Students represent equations given in rectangular coordinates in terms of polar coordinates.
    • 17.0 Students are familiar with complex numbers. They can represent a complex number in polar form and know how to multiply complex numbers in their polar form.

Secondary

  • AP Calculus

    • APC.08 The student will apply the derivative to solve problems. This will include analysis of curves and the ideas of concavity and monotonicity; optimization involving global and local extrema; modeling of rates of change and related rates; use of implicit differentiation to find the derivative of an inverse function; interpretation of the derivative as a rate of change in applied contexts, including velocity, speed, and acceleration; and, differentiation of nonlogarithmic functions, using the technique of logarithmic differentiation. * * AP Calculus BC will also apply the derivative to solve problems. This will include analysis of planar curves given in parametric form, polar form, and vector form, including velocity and acceleration vectors; numerical solution of differential equations, using Euler's method; l'Hopital's Rule to test the convergence of improper integrals and series; and, geometric interpretation of differential equations via slope fields and the relationship between slope fields and the solution curves for the differential equations.
    • APC.09 The student will apply formulas to find derivatives. This will include derivatives of algebraic, trigonometric, exponential, logarithmic, and inverse trigonometric functions; derivations of sums, products, quotients, inverses, and composites (chain rule) of elementary functions; derivatives of implicitly defined functions; and, higher order derivatives of algebraic, trigonometric, exponential, and logarithmic, functions. * * AP Calculus BC will also include finding derivatives of parametric, polar, and vector functions.
    • APC.15 The student will use integration techniques and appropriate integrals to model physical, biological, and economic situations. The emphasis will be on using the integral of a rate of change to give accumulated change or on using the method of setting up an approximating Riemann sum and representing its limit as a definite integral. Specific applications will include the area of a region; the volume of a solid with known cross-section; the average value of a function; and, the distance traveled by a particle along a line. * * AP Calculus BC will include finding the area of a region (including a region bounded by polar curves) and finding the length of a curve (including a curve given in parametric form).

  • Mathematical Analysis

    • MA.10 The student will investigate and identify the characteristics of the graphs of polar equations, using graphing utilities. This will include classification of polar equations, the effects of changes in the parameters in polar equations, conversion of complex numbers from rectangular form to polar form and vice versa, and the intersection of the graphs of polar equations.

Student Prerequisites

  • Mathematics: Students must be able to:
    • Evaluate trigonometric functions
    • Plot points in polar coordinates
    • Understand the unit circle
  • Technological: Students must be able to:
    • Perform basic mouse manipulations such as point, click, and drag
    • Use a browser for experimenting with the activities

Teacher Preparation

Students will need:

Key Terms

Cardioid

The resulting graph in polar coordinates of a function of the form a+b*sin(t) or a+b*cos(t) where |a| = |b|

Limacon

The resulting graph in polar coordinates of a function of the form a+b*sin(t) or a+b*cos(t) where |a| ≠ |b|

Petal Curve

See Rose Curve

Rose Curve

The graph of a function in polar coordinates of the form a*sin(b*t) or a*cos(b*t) where a ≠ 0 and b is an integer > 1

Lesson Outline

  1. Focus and Review

    • Review process of sketching other graphs
      • Choose x values
      • Evaluate f(x) for those values
      • Plot ordered pairs (x, f(x))
    • Lead the class in a brief discussion on polar coordinates
    • Review unit circle
      • Show students a blank unit circle. Point to different places and say something like:
        • About how many degrees is it to here? Radians to here? Sine of that angle? Cosine of that angle?

  2. Objectives

    Let the students know what they will be doing and learning today. Say something like this:

    • Today, we are going to learn about combining trigonometry and polar coordinates and the parent graphs created by trigonometric functions in polar coordinates.
    • We are going to use the computers to do this, but please do not turn your computers on until I ask you to. I want to show you a little about this activity first.

  3. Teacher Input

    Lead a discussion on Plotting Functions in the Polar Coordinate System

  4. Guided Practice

    • Guide students through the first question on the Polar Coordinates Exploration Questions
    • Tell students to go to the Polar Coordinates Activity, plot the points they calculated, then enter in the function to see if they were correct.
    • Encourage students to slightly change the previous function and observe their results
    • Have a prepared picture of a Limacon, Cardioid, and petal curve.
      • Hold up one of your graphs and say something like:
        • Does anyone's new graph look something like this? What is the function that gave you this graph?
      • Then introduce the name of that curve and write it on the board with the sample equation the student used.
      • If you present one of the graphs and none of the students has a graph like it then give students a little more time to try and make a graph that represents the one you are showing.
    • For one example, have students plot points with theta values greater than 2*pi.
      • Have them graph these points and note whether or not they are on the same graph.
      • When they notice that these points are already on the graph, make a note about the cyclic nature of sine and cosine. Say something like:
        • Notice that after a certain point, the r values start to repeat. This is due to the cyclic nature of sine and cosine.
        • The first point where the r values start to repeat is what we call the period of the function.
        • No matter what, after 2*pi, the r values will start to repeat.

  5. Independent Practice

    • Have students complete the rest of the questions on the Polar Coordinates Exploration Questions.
    • Prompt students with various graphs and have them write down a possible function for that graph.

  6. Closure

    • Bring students back together for a discussion of their findings. Have a few graphs on hand that you can refer to and ask questions about. Say something like this:
      • Which of the three terms we learned today best describes this graph? (i.e. cardioid, limacon, petal curve)
      • What might be a possible equation for this graph?
        • Why did you choose sine/cosine?
        • Why did you choose those values of a and b?

Alternate Outline

If time is a constraint, this lesson can be truncated by removing the exploration part of the activity.

  • Start the lesson the same way, by having the students sketch one graph by hand and check it on the computer.
  • Then introduce the other types of graphs they can attain with trigonometric functions, i.e. cardioids and limacons, as well as the equations that will give them these graphs.
  • Then hold up the graphs that you have prepared and have the class come up with a possible equation for each graph.
  • Make them defend why they chose either sine or cosine and the values of their constants.

Suggested Follow-Up

After these discussions and activities, students will have learned about graphing in the polar coordinate plane and be able to identify graphs of trigonometric functions in the polar coordinate plane. Suggested follow-up could include transforming Cartesian coordinates into polar coordinates.