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7.1: Polar Necessities

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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 $b < a$.

Feb 23, 2012

Aug 19, 2014