1.
- Graph [latex]r=k\text{,}[/latex] for [latex]k=1, 2, 3\text{.}[/latex] How does the graph change for different values of [latex]k\text{?}[/latex]
- Write a Cartesian equation for each graph in part (a).
Chapter 10: Polar Coordinates and Complex Numbers
Practice each skill in the Homework Problems listed.
In Problems 1-4, use your calculator to graph the equations.
Complete the table of values for each equation. Plot the points in order of increasing [latex]\theta\text{.}[/latex] What is different about the two graphs? Equation 1: [latex]~~r=2[/latex]
| [latex]\theta[/latex] | [latex]0[/latex] | [latex]\dfrac{\pi}{4}[/latex] | [latex]\dfrac{\pi}{2}[/latex] | [latex]\dfrac{3\pi}{4}[/latex] | [latex]\pi[/latex] | [latex]\dfrac{5\pi}{4}[/latex] | [latex]\dfrac{3\pi}{2}[/latex] | [latex]\dfrac{7\pi}{4}[/latex] |
| [latex]r=2[/latex] | [latex]\hphantom{000}[/latex] | [latex]\hphantom{000}[/latex] | [latex]\hphantom{000}[/latex] | [latex]\hphantom{000}[/latex] | [latex]\hphantom{000}[/latex] | [latex]\hphantom{000}[/latex] | [latex]\hphantom{000}[/latex] | [latex]\hphantom{000}[/latex] |
Equation 2: [latex]~~r=-2[/latex]
| [latex]\theta[/latex] | [latex]0[/latex] | [latex]\dfrac{\pi}{4}[/latex] | [latex]\dfrac{\pi}{2}[/latex] | [latex]\dfrac{3\pi}{4}[/latex] | [latex]\pi[/latex] | [latex]\dfrac{5\pi}{4}[/latex] | [latex]\dfrac{3\pi}{2}[/latex] | [latex]\dfrac{7\pi}{4}[/latex] |
| [latex]r=-2[/latex] | [latex]\hphantom{000}[/latex] | [latex]\hphantom{000}[/latex] | [latex]\hphantom{000}[/latex] | [latex]\hphantom{000}[/latex] | [latex]\hphantom{000}[/latex] | [latex]\hphantom{000}[/latex] | [latex]\hphantom{000}[/latex] | [latex]\hphantom{000}[/latex] |
Graph each line and label the points with their coordinates. How are the points on the two lines related? Equation 1: [latex]~~\theta = \dfrac{\pi}{4}[/latex]
| [latex]\theta = \dfrac{\pi}{4}[/latex] | [latex]\dfrac{\pi}{4}[/latex] | [latex]\dfrac{\pi}{4}[/latex] | [latex]\dfrac{\pi}{4}[/latex] | [latex]\dfrac{\pi}{4}[/latex] | [latex]\dfrac{\pi}{4}[/latex] |
| [latex]r[/latex] | [latex]-2[/latex] | [latex]-1[/latex] | [latex]0[/latex] | [latex]1[/latex] | [latex]2[/latex] |
Equation 1: [latex]~~\theta = \dfrac{5\pi}{4}[/latex]
| [latex]\theta = \dfrac{5\pi}{4}[/latex] | [latex]\dfrac{5\pi}{4}[/latex] | [latex]\dfrac{5\pi}{4}[/latex] | [latex]\dfrac{5\pi}{4}[/latex] | [latex]\dfrac{5\pi}{4}[/latex] | [latex]\dfrac{5\pi}{4}[/latex] |
| [latex]r[/latex] | [latex]-2[/latex] | [latex]-1[/latex] | [latex]0[/latex] | [latex]1[/latex] | [latex]2[/latex] |
| [latex]\theta[/latex] | [latex]\pi[/latex] | [latex]\dfrac{5\pi}{4}[/latex] | [latex]\dfrac{3\pi}{2}[/latex] | [latex]\dfrac{7\pi}{4}[/latex] | [latex]2\pi[/latex] |
| [latex]r[/latex] | [latex]\hphantom{000}[/latex] | [latex]\hphantom{000}[/latex] | [latex]\hphantom{000}[/latex] | [latex]\hphantom{000}[/latex] | [latex]\hphantom{000}[/latex] |
| [latex]\theta[/latex] | [latex]\pi[/latex] | [latex]\dfrac{5\pi}{4}[/latex] | [latex]\dfrac{3\pi}{2}[/latex] | [latex]\dfrac{7\pi}{4}[/latex] | [latex]2\pi[/latex] |
| [latex]r[/latex] | [latex]\hphantom{000}[/latex] | [latex]\hphantom{000}[/latex] | [latex]\hphantom{000}[/latex] | [latex]\hphantom{000}[/latex] | [latex]\hphantom{000}[/latex] |
Complete the table of values for each cardioid and graph the equation.
| [latex]\theta[/latex] | [latex]0[/latex] | [latex]\dfrac{\pi}{2}[/latex] | [latex]\pi[/latex] | [latex]\dfrac{3\pi}{2}[/latex] | [latex]2\pi[/latex] |
| [latex]r[/latex] | [latex]\hphantom{000}[/latex] | [latex]\hphantom{000}[/latex] | [latex]\hphantom{000}[/latex] | [latex]\hphantom{000}[/latex] | [latex]\hphantom{000}[/latex] |
Complete the table of values for each cardioid and graph the equation.
| [latex]\theta[/latex] | [latex]0[/latex] | [latex]\dfrac{\pi}{2}[/latex] | [latex]\pi[/latex] | [latex]\dfrac{3\pi}{2}[/latex] | [latex]2\pi[/latex] |
| [latex]r[/latex] | [latex]\hphantom{000}[/latex] | [latex]\hphantom{000}[/latex] | [latex]\hphantom{000}[/latex] | [latex]\hphantom{000}[/latex] | [latex]\hphantom{000}[/latex] |
Complete the table of values for each limaçon and graph the equation.
| [latex]\theta[/latex] | [latex]0[/latex] | [latex]\dfrac{\pi}{2}[/latex] | [latex]\pi[/latex] | [latex]\dfrac{3\pi}{2}[/latex] | [latex]2\pi[/latex] |
| [latex]r[/latex] | [latex]\hphantom{000}[/latex] | [latex]\hphantom{000}[/latex] | [latex]\hphantom{000}[/latex] | [latex]\hphantom{000}[/latex] | [latex]\hphantom{000}[/latex] |
Complete the table of values for each limaçon and graph the equation.
| [latex]\theta[/latex] | [latex]0[/latex] | [latex]\dfrac{\pi}{2}[/latex] | [latex]\pi[/latex] | [latex]\dfrac{3\pi}{2}[/latex] | [latex]2\pi[/latex] |
| [latex]r[/latex] | [latex]\hphantom{000}[/latex] | [latex]\hphantom{000}[/latex] | [latex]\hphantom{000}[/latex] | [latex]\hphantom{000}[/latex] | [latex]\hphantom{000}[/latex] |
Graph the Archimedean spiral [latex]r=\theta\text{.}[/latex] Set your window to
[latex][latex] \begin{aligned}[t] \theta\text{min}=0~~~~~~~~~~\theta\text{max}=8\pi\\ \text{Xmin}=-20~~~~\text{Xmax}=20\\ \text{Ymin}=-20~~~~\text{Ymax}=20\\ \end{aligned}[/latex]
Then graph by pressing Zoom 5.
Graph the logarithmic spiral [latex]r=e^{0.2\theta}\text{.}[/latex] Set your window to
[latex][latex] \begin{aligned}[t] \theta\text{min}=0~~~~~~~~~~\theta\text{max}=8\pi\\ \text{Xmin}=-100~~~~\text{Xmax}=100\\ \text{Ymin}=-100~~~~\text{Ymax}=100\\ \end{aligned}[/latex]
Then graph by pressing Zoom 5.
| [latex]\theta[/latex] | [latex]\hphantom{000}[/latex] | [latex]\hphantom{000}[/latex] | [latex]\hphantom{000}[/latex] | [latex]\hphantom{000}[/latex] | [latex]\hphantom{000}[/latex] | [latex]\hphantom{000}[/latex] | [latex]\hphantom{000}[/latex] | [latex]\hphantom{000}[/latex] | [latex]\hphantom{000}[/latex] |
| [latex]3\theta[/latex] | [latex]0[/latex] | [latex]\dfrac{\pi}{4}[/latex] | [latex]\dfrac{\pi}{2}[/latex] | [latex]\dfrac{3\pi}{4}[/latex] | [latex]\pi[/latex] | [latex]\dfrac{5\pi}{4}[/latex] | [latex]\dfrac{3\pi}{2}[/latex] | [latex]\dfrac{7\pi}{4}[/latex] | [latex]2\pi[/latex] |
| [latex]y[/latex] | [latex]\hphantom{000}[/latex] | [latex]\hphantom{000}[/latex] | [latex]\hphantom{000}[/latex] | [latex]\hphantom{000}[/latex] | [latex]\hphantom{000}[/latex] | [latex]\hphantom{000}[/latex] | [latex]\hphantom{000}[/latex] | [latex]\hphantom{000}[/latex] | [latex]\hphantom{000}[/latex] |
Complete the table and graph the equation [latex]r=\sin 3\theta[/latex] in polar coordinates for [latex]0 \le \theta\le 2\pi\text{.}[/latex]
| [latex]\theta[/latex] | [latex]\hphantom{000}[/latex] | [latex]\hphantom{000}[/latex] | [latex]\hphantom{000}[/latex] | [latex]\hphantom{000}[/latex] | [latex]\hphantom{000}[/latex] | [latex]\hphantom{000}[/latex] | [latex]\hphantom{000}[/latex] | [latex]\hphantom{000}[/latex] | [latex]\hphantom{000}[/latex] |
| [latex]3\theta[/latex] | [latex]0[/latex] | [latex]\dfrac{\pi}{4}[/latex] | [latex]\dfrac{\pi}{2}[/latex] | [latex]\dfrac{3\pi}{4}[/latex] | [latex]\pi[/latex] | [latex]\dfrac{5\pi}{4}[/latex] | [latex]\dfrac{3\pi}{2}[/latex] | [latex]\dfrac{7\pi}{4}[/latex] | [latex]2\pi[/latex] |
| [latex]r[/latex] | [latex]\hphantom{000}[/latex] | [latex]\hphantom{000}[/latex] | [latex]\hphantom{000}[/latex] | [latex]\hphantom{000}[/latex] | [latex]\hphantom{000}[/latex] | [latex]\hphantom{000}[/latex] | [latex]\hphantom{000}[/latex] | [latex]\hphantom{000}[/latex] | [latex]\hphantom{000}[/latex] |
| [latex]\theta[/latex] | [latex]\hphantom{000}[/latex] | [latex]\hphantom{000}[/latex] | [latex]\hphantom{000}[/latex] | [latex]\hphantom{000}[/latex] | [latex]\hphantom{000}[/latex] | [latex]\hphantom{000}[/latex] | [latex]\hphantom{000}[/latex] | [latex]\hphantom{000}[/latex] | [latex]\hphantom{000}[/latex] |
| [latex]2\theta[/latex] | [latex]0[/latex] | [latex]\dfrac{\pi}{4}[/latex] | [latex]\dfrac{\pi}{2}[/latex] | [latex]\dfrac{3\pi}{4}[/latex] | [latex]\pi[/latex] | [latex]\dfrac{5\pi}{4}[/latex] | [latex]\dfrac{3\pi}{2}[/latex] | [latex]\dfrac{7\pi}{4}[/latex] | [latex]2\pi[/latex] |
| [latex]y[/latex] | [latex]\hphantom{000}[/latex] | [latex]\hphantom{000}[/latex] | [latex]\hphantom{000}[/latex] | [latex]\hphantom{000}[/latex] | [latex]\hphantom{000}[/latex] | [latex]\hphantom{000}[/latex] | [latex]\hphantom{000}[/latex] | [latex]\hphantom{000}[/latex] | [latex]\hphantom{000}[/latex] |
Complete the table and graph the equation [latex]r=\cos 2\theta[/latex] in polar coordinates for [latex]0 \le \theta\le 2\pi\text{.}[/latex]
| [latex]\theta[/latex] | [latex]\hphantom{000}[/latex] | [latex]\hphantom{000}[/latex] | [latex]\hphantom{000}[/latex] | [latex]\hphantom{000}[/latex] | [latex]\hphantom{000}[/latex] | [latex]\hphantom{000}[/latex] | [latex]\hphantom{000}[/latex] | [latex]\hphantom{000}[/latex] | [latex]\hphantom{000}[/latex] |
| [latex]2\theta[/latex] | [latex]0[/latex] | [latex]\dfrac{\pi}{4}[/latex] | [latex]\dfrac{\pi}{2}[/latex] | [latex]\dfrac{3\pi}{4}[/latex] | [latex]\pi[/latex] | [latex]\dfrac{5\pi}{4}[/latex] | [latex]\dfrac{3\pi}{2}[/latex] | [latex]\dfrac{7\pi}{4}[/latex] | [latex]2\pi[/latex] |
| [latex]r[/latex] | [latex]\hphantom{000}[/latex] | [latex]\hphantom{000}[/latex] | [latex]\hphantom{000}[/latex] | [latex]\hphantom{000}[/latex] | [latex]\hphantom{000}[/latex] | [latex]\hphantom{000}[/latex] | [latex]\hphantom{000}[/latex] | [latex]\hphantom{000}[/latex] | [latex]\hphantom{000}[/latex] |
| [latex]\theta[/latex] | [latex]0[/latex] | [latex]\dfrac{\pi}{4}[/latex] | [latex]\dfrac{\pi}{2}[/latex] | [latex]\dfrac{3\pi}{4}[/latex] | [latex]\pi[/latex] | [latex]\dfrac{5\pi}{4}[/latex] | [latex]\dfrac{3\pi}{2}[/latex] | [latex]\dfrac{7\pi}{4}[/latex] | [latex]2\pi[/latex] |
| [latex]y[/latex] | [latex]\hphantom{000}[/latex] | [latex]\hphantom{000}[/latex] | [latex]\hphantom{000}[/latex] | [latex]\hphantom{000}[/latex] | [latex]\hphantom{000}[/latex] | [latex]\hphantom{000}[/latex] | [latex]\hphantom{000}[/latex] | [latex]\hphantom{000}[/latex] | [latex]\hphantom{000}[/latex] |
Complete the table and graph the equation [latex]r=2+2\cos\theta[/latex] in polar coordinates for [latex]0 \le \theta\le 2\pi\text{.}[/latex]
| [latex]\theta[/latex] | [latex]0[/latex] | [latex]\dfrac{\pi}{4}[/latex] | [latex]\dfrac{\pi}{2}[/latex] | [latex]\dfrac{3\pi}{4}[/latex] | [latex]\pi[/latex] | [latex]\dfrac{5\pi}{4}[/latex] | [latex]\dfrac{3\pi}{2}[/latex] | [latex]\dfrac{7\pi}{4}[/latex] | [latex]2\pi[/latex] |
| [latex]r[/latex] | [latex]\hphantom{000}[/latex] | [latex]\hphantom{000}[/latex] | [latex]\hphantom{000}[/latex] | [latex]\hphantom{000}[/latex] | [latex]\hphantom{000}[/latex] | [latex]\hphantom{000}[/latex] | [latex]\hphantom{000}[/latex] | [latex]\hphantom{000}[/latex] | [latex]\hphantom{000}[/latex] |
| [latex]\theta[/latex] | [latex]0[/latex] | [latex]\dfrac{\pi}{4}[/latex] | [latex]\dfrac{\pi}{2}[/latex] | [latex]\dfrac{3\pi}{4}[/latex] | [latex]\pi[/latex] | [latex]\dfrac{5\pi}{4}[/latex] | [latex]\dfrac{3\pi}{2}[/latex] | [latex]\dfrac{7\pi}{4}[/latex] | [latex]2\pi[/latex] |
| [latex]y[/latex] | [latex]\hphantom{000}[/latex] | [latex]\hphantom{000}[/latex] | [latex]\hphantom{000}[/latex] | [latex]\hphantom{000}[/latex] | [latex]\hphantom{000}[/latex] | [latex]\hphantom{000}[/latex] | [latex]\hphantom{000}[/latex] | [latex]\hphantom{000}[/latex] | [latex]\hphantom{000}[/latex] |
Complete the table and graph the equation [latex]r=1-\sin\theta[/latex] in polar coordinates for [latex]0 \le \theta\le 2\pi\text{.}[/latex]
| [latex]\theta[/latex] | [latex]0[/latex] | [latex]\dfrac{\pi}{4}[/latex] | [latex]\dfrac{\pi}{2}[/latex] | [latex]\dfrac{3\pi}{4}[/latex] | [latex]\pi[/latex] | [latex]\dfrac{5\pi}{4}[/latex] | [latex]\dfrac{3\pi}{2}[/latex] | [latex]\dfrac{7\pi}{4}[/latex] | [latex]2\pi[/latex] |
| [latex]r[/latex] | [latex]\hphantom{000}[/latex] | [latex]\hphantom{000}[/latex] | [latex]\hphantom{000}[/latex] | [latex]\hphantom{000}[/latex] | [latex]\hphantom{000}[/latex] | [latex]\hphantom{000}[/latex] | [latex]\hphantom{000}[/latex] | [latex]\hphantom{000}[/latex] | [latex]\hphantom{000}[/latex] |
For Problems 25–42, use the catalog of polar graphs to help you identify and sketch the following curves. Check your work by graphing with a calculator.
[latex]r=3\cos\theta[/latex]
[latex]r=2\sin \theta[/latex]
[latex]\theta=\dfrac{\pi}{4}[/latex]
[latex]\theta=\dfrac{4\pi}{3}[/latex]
[latex]r=4[/latex]
[latex]r=2[/latex]
[latex]r=2+2\sin \theta[/latex]
[latex]r=3+3\cos\theta[/latex]
[latex]r=2-\cos\theta[/latex]
[latex]r=1-3\sin \theta[/latex]
[latex]r=3\sin 2\theta[/latex]
[latex]r=2\cos 3\theta[/latex]
[latex]r=2\cos 5\theta[/latex]
[latex]r=4\sin 4\theta[/latex]
[latex]r=2+3\sin \theta[/latex]
[latex]r=3+2\sin \theta[/latex]
[latex]r^2=\cos 2\theta[/latex]
[latex]r^2=4\sin2\theta[/latex]
For Problems 43–52, identify each curve and graph it.
[latex]r\csc\theta = 2[/latex]
[latex]r=2\sec \theta[/latex]
[latex]r^2=4,~ 0 \le \theta \le \dfrac{3\pi}{4}[/latex]
[latex]\theta = \dfrac{\pi}{4},~ {|r|} \lt 2[/latex]
[latex]r=\sin \theta,~ \dfrac{3\pi}{4} \le \theta \le \dfrac{5\pi}{4}[/latex]
[latex]r=\cos \theta,~ 0 \le \theta \le \dfrac{\pi}{2}[/latex]
[latex]r=2\sin 2\theta \cos 2\theta[/latex]
[latex]r=\cos^2 \theta - \sin^2 \theta[/latex]
[latex]r(1-\cos\theta)=\sin^2\theta[/latex]
[latex]r \sec \theta=\sec\theta - \tan \theta[/latex]
For Problems 53–58, graph the following polar curves. Do you recognize them?
[latex]r = \dfrac{2}{1-\cos\theta}[/latex]
[latex]r = \dfrac{6}{2+\sin\theta}[/latex]
[latex]r = \dfrac{2}{2-\cos\theta}[/latex]
[latex]r = \dfrac{1}{1+\sin\theta}[/latex]
[latex]r = \dfrac{1}{1+2\sin\theta}[/latex]
[latex]r = \dfrac{3}{2-3\cos\theta}[/latex]
For Problems 59–66, write a polar equation for the graph.
For Problems 67–74, find the coordinates of the intersection points of the two curves analytically. Then graph the curves to verify your answers.
[latex]r=\cos\theta,~ r=1-\cos\theta[/latex]
[latex]r=\sin\theta,~ r=\cos\theta[/latex]
[latex]r=3\sin\theta,~ r=3\cos\theta[/latex]
[latex]r=\sin 2\theta,~ r=\cos 2\theta[/latex]
[latex]r=1,~ r=1-\cos\theta[/latex]
[latex]r=3\cos\theta,~ r=1+\cos\theta[/latex]
[latex]r=2+\sin\theta,~ r=2-\cos\theta[/latex]
[latex]r=\sin\theta,~ r=\sin 2\theta[/latex]
For Problems 75–82, graph the polar curve.
[latex]r^2 = \tan \theta[/latex]
[latex]r^2 = \cot \theta[/latex]
[latex]r=\csc\theta-2[/latex] (conchoid)
[latex]r=\tan\theta[/latex] (kappa curve)
[latex]r=\cos 2\theta \sec\theta[/latex] (strophoid)
[latex]r=\sin \theta \tan \theta[/latex] (cissoid)
[latex]r=\dfrac{1}{\sqrt{\theta}}[/latex]
[latex]r=\cos\ \dfrac{\theta}{2},~ 0 \le \theta \le 4\pi[/latex]
Graph the polar curves [latex]r=1-2\sin n\theta[/latex] for [latex]n=2,3,4,5,6\text{.}[/latex] Explain how the value of the parameter [latex]n[/latex] affects the curve.
Graph the polar curves [latex]r=1-3\cos n\theta[/latex] for [latex]n=2,3,4,5,6\text{.}[/latex] Explain how the value of the parameter [latex]n[/latex] affects the curve.