7.1: Polar Necessities
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
- Student Worksheet: Trigonometric Patterns http://www.ck12.org/flexr/chapter/9704, scroll down to the third activity.
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 \begin{align*}\left(- \frac{\pi}{3}, 3 \right)\end{align*}?
- What about if r was negative? For example, move to \begin{align*}\left(\frac{\pi}{2}, -6 \right)\end{align*}.
Solutions
1. See image below.
2. If \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 \begin{align*}r = a\end{align*}, where \begin{align*}a\end{align*} is a constant.
A polar rose with even petals is in the form \begin{align*}r(\theta) = a \cdot \sin(n \theta)\end{align*}, where \begin{align*}n\end{align*} is even.
A polar rose with odd petals is in the form \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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