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# 7.1: Polar Necessities

Difficulty Level: At Grade Created by: CK-12

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 (π3,3)\begin{align*}\left(- \frac{\pi}{3}, 3 \right)\end{align*}?
• What about if r was negative? For example, move to (π2,6)\begin{align*}\left(\frac{\pi}{2}, -6 \right)\end{align*}.

## Solutions

1. See image below.

2. If r(θ)=cos(θ),r(π3)=0.5\begin{align*}r(\theta) = \cos(\theta), r \left(\frac{\pi}{3} \right) = 0.5\end{align*}.

3. a heart or cardioid

4. A circle is in the form r=a\begin{align*}r = a\end{align*}, where a\begin{align*}a\end{align*} is a constant.

A polar rose with even petals is in the form r(θ)=asin(nθ)\begin{align*}r(\theta) = a \cdot \sin(n \theta)\end{align*}, where n\begin{align*}n\end{align*} is even.

A polar rose with odd petals is in the form r(θ)=asin(nθ)\begin{align*}r(\theta) = a \cdot \sin(n \theta)\end{align*}, where \begin{align*}n\end{align*} is odd.

A limaçon with an inner loop comes form \begin{align*}r(\theta) = b + a \cdot \cos(\theta)\end{align*}, where \begin{align*}b < a\end{align*}.

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