Polar Coordinates

Author: John J Weber III, PhD Corresponding Textbook Sections:

• Section 10.3 – Polar Coordinates

Expected Educational Results

• Objective 16–01: I can convert a point between polar coordinates and cartesian coordinates.

• Objective 16–02: I can convert equations between polar coordinates and cartesian coordinates.

• Objective 16–03: I can use technology to sketch a polar curve.

• Objective 16–04: I can find the derivative of a polar curve.

• Objective 16–05: I can use the derivative of a polar curve to find the equation of the tangent line to the curve, points where the curve has horizontal tangent lines, and points where the curve has vertical tangent lines.

Bloom’s Taxonomy

A modern version of Bloom’s Taxonomy is included here to recognize various different levels of understanding and to encourage you to work towards higher-order understanding (those at the top of the pyramid). All Objectives, Investigations, Activities, etc. are color-coded with the level of understanding.

Polar Coordinate System

Sketching Polar Curves

NOTE: You are only expected to graph polar curves using technology.

Investigation 03

Use the embedded DESMOS graph to sketch the following graphs:

NOTE: In DESMOS, when you use the variables $r$$r$ and $\theta$$\theta$, DESMOS creates a polar graph. To type $\theta$$\theta$, either use the display keypad [located on bottom-left of window] or type theta.

1. ${\displaystyle r=\theta }$$\displaystyle r=\theta$

2. ${\displaystyle r=\cos (3\theta )}$$\displaystyle r=\cos{(3\theta)}$

3. ${\displaystyle r=\cos (4\theta )}$$\displaystyle r=\cos{(4\theta)}$

4. ${\displaystyle r=2\sec (\theta )}$$\displaystyle r=2\sec{(\theta)}$

5. ${\displaystyle r=3\sin (\theta )}$$\displaystyle r=3\sin{(\theta)}$

6. ${\displaystyle r=1+2\cos (\theta )}$$\displaystyle r=1+2\cos{(\theta)}$

7. ${\displaystyle r=3+2\sin (\theta )}$$\displaystyle r=3+2\sin{(\theta)}$

8. ${\displaystyle r=2-2\sin (\theta )}$$\displaystyle r=2-2\sin{(\theta)}$

9. ${\displaystyle r^2=2\sin (2\theta )}$$\displaystyle r^2=2\sin{(2\theta)}$

10. ${\displaystyle r=\frac{1}{\sin (\theta )-\cos (\theta )}}$$\displaystyle r=\frac{1}{\sin{(\theta)}-\cos{(\theta)}}$

11. ${\displaystyle r=8+3\csc (\theta )}$$\displaystyle r=8+3\csc{(\theta)}$

12. ${\displaystyle r=\sin (\theta )+{(\sin \left(\frac{5\theta }{2}\right))}^3}$$\displaystyle r=\sin{(\theta)}+\left(\sin{\left(\frac{5\theta}{2}\right)}\right)^3$

13. ${\displaystyle r={(\cos (5\theta ))}^2+\sin (3\theta )+0.3}$$\displaystyle r=\left(\cos{(5\theta)}\right)^2+\sin{(3\theta)}+0.3$

Use Technology to Graph Polar Curves

Mathematica

NOTE: In Mathematica, if you want to use the variable $\theta$$\theta$, you will have to find $\theta$$\theta$ in a menu; a simpler method is to replace $\theta$$\theta$ with $t$$t$.

1
(* The graph of r=4 cos(2cos(t)) *)
2
PolarPlot[4 Cos[2 Cos[t]],{t, 0 ,2 Pi}]

Warnings:

1. Be very careful with the syntax. Syntax is the set of rules on how to write computer code. Every software program has its own unique syntax. Some basic Mathematica syntax is located at: http://www.jjw3.com/TECH_Common_Functions.pdf.

2. To execute code (including comment codes), press and hold the SHIFT key and press the ENTER key.

Homework

At this time, you should be able to complete the following assignments:

• Section 10.3: # 29, 31, 33, 35, 37, 39, 41, 43, 45, 47, 54.

CC BY-NC-SA 4.0

This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License [http://creativecommons.org/licenses/by-nc-sa/4.0/].

Last Modified: Tuesday, 3 November 2020 6:51 EDT