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This activity is intended to supplement Trigonometry, Chapter 6, Lesson 4.

ID: 12558

Time Required: 15 minutes

Activity Overview

Students will explore what is necessary to understand the calculus of polar equations. Students will graphically and algebraically find the slope of the tangent line at a point on a polar graph. Finding the area of a region of a polar curve will be determined using the area formula.

Topic: Polar Equations

  • Find the slope of a polar equation at a particular point.
  • Find the area of polar equation.

Teacher Preparation and Notes

  • Make sure each students' calculator is in RADIANS (RAD) and POLAR (POL) in the MODE menu.

Associated Materials

Plotting Coordinates & Exploring Polar Graphs

Students begin the activity by plotting points on a polar graph. This should be a refresher of polar coordinates for most students. Students practice using the calculator to graph a polar equation.

Discussion Questions

  • What do you think it means to have a negative angle, like \left(- \frac{\pi}{3}, 3 \right)?
  • What about if r was negative? For example, move to \left(\frac{\pi}{2}, -6 \right).

Solutions

1. See image below.

2. If r(\theta) = \cos(\theta), r \left(\frac{\pi}{3} \right)  = 0.5.

3. a heart or cardioid

4. A circle is in the form r = a, where a is a constant.

A polar rose with even petals is in the form r(\theta) = a \cdot \sin(n \theta), where n is even.

A polar rose with odd petals is in the form r(\theta) = a \cdot \sin(n \theta), where n is odd.

A limaçon with an inner loop comes form r(\theta) = b + a \cdot \cos(\theta), where b < a.

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Date Created:

Feb 23, 2012

Last Modified:

Aug 19, 2014
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