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Determine the symmetry of the graph determined by the equation $r=2\phantom{\rule{0.2em}{0ex}}\text{cos}\left(3\theta \right)$ and create a graph.
Symmetric with respect to the polar axis.
In the following exercises, plot the point whose polar coordinates are given by first constructing the angle $\theta $ and then marking off the distance r along the ray.
$\left(\mathrm{-2},\frac{5\pi}{3}\right)$
$\left(\mathrm{-4},\frac{3\pi}{4}\right)$
$\left(2,\frac{5\pi}{6}\right)$
For the following exercises, consider the polar graph below. Give two sets of polar coordinates for each point.
Coordinates of point A .
Coordinates of point B .
$B\begin{array}{cc}\left(3,\frac{\text{\u2212}\pi}{3}\right)\hfill & B\left(\mathrm{-3},\frac{2\pi}{3}\right)\hfill \end{array}$
Coordinates of point C .
Coordinates of point D .
$D\left(5,\frac{7\pi}{6}\right)D\left(\mathrm{-5},\frac{\pi}{6}\right)$
For the following exercises, the rectangular coordinates of a point are given. Find two sets of polar coordinates for the point in $\left(0,2\pi \right].$ Round to three decimal places.
$\left(2,\phantom{\rule{0.2em}{0ex}}2\right)$
$\left(3,\mathrm{-4}\right)$ (3, −4)
$\begin{array}{cc}\left(5,\mathrm{-0.927}\right)\hfill & \left(\mathrm{-5},\mathrm{-0.927}+\pi \right)\hfill \end{array}$
$\left(8,\phantom{\rule{0.2em}{0ex}}15\right)$
$\left(\mathrm{-6},\phantom{\rule{0.2em}{0ex}}8\right)$
$\left(10,\mathrm{-0.927}\right)\left(\mathrm{-10},\mathrm{-0.927}+\pi \right)$
$\left(4,\phantom{\rule{0.2em}{0ex}}3\right)$
$\left(3,\text{\u2212}\sqrt{3}\right)$
$\left(2\sqrt{3},\mathrm{-0.524}\right)\left(\mathrm{-2}\sqrt{3},\mathrm{-0.524}+\pi \right)$
For the following exercises, find rectangular coordinates for the given point in polar coordinates.
$\left(2,\frac{5\pi}{4}\right)$
$\left(\mathrm{-2},\frac{\pi}{6}\right)$
$\left(\begin{array}{cc}\text{\u2212}\sqrt{3},\hfill & \mathrm{-1}\hfill \end{array}\right)$
$\left(5,\frac{\pi}{3}\right)$
$\left(1,\frac{7\pi}{6}\right)$
$\left(\begin{array}{cc}-\frac{\sqrt{3}}{2},\hfill & \frac{\mathrm{-1}}{2}\hfill \end{array}\right)$
$\left(\mathrm{-3},\frac{3\pi}{4}\right)$
$\left(0,\frac{\pi}{2}\right)$
$\left(\begin{array}{cc}0,\hfill & 0\hfill \end{array}\right)$
$\left(\mathrm{-4.5},6.5\right)$
For the following exercises, determine whether the graphs of the polar equation are symmetric with respect to the $x$ -axis, the $y$ -axis, or the origin.
$r=3\phantom{\rule{0.2em}{0ex}}\text{sin}(2\theta )$
Symmetry with respect to the x -axis, y -axis, and origin.
${r}^{2}=9\phantom{\rule{0.2em}{0ex}}\text{cos}\phantom{\rule{0.2em}{0ex}}\theta $
$r=\text{cos}\left(\frac{\theta}{5}\right)$
Symmetric with respect to x -axis only.
$r=2\phantom{\rule{0.2em}{0ex}}\text{sec}\phantom{\rule{0.2em}{0ex}}\theta $
$r=1+\text{cos}\phantom{\rule{0.2em}{0ex}}\theta $
Symmetry with respect to x -axis only.
For the following exercises, describe the graph of each polar equation. Confirm each description by converting into a rectangular equation.
$r=\text{sec}\phantom{\rule{0.2em}{0ex}}\theta $
For the following exercises, convert the rectangular equation to polar form and sketch its graph.
${x}^{2}+{y}^{2}=16$
${x}^{2}-{y}^{2}=16$
Hyperbola; polar form
${r}^{2}\text{cos}(2\theta )=16$ or
${r}^{2}=16\phantom{\rule{0.2em}{0ex}}\text{sec}\phantom{\rule{0.2em}{0ex}}\theta .$
For the following exercises, convert the rectangular equation to polar form and sketch its graph.
$3x-y=2$
$r=\frac{2}{3\phantom{\rule{0.2em}{0ex}}\text{cos}\phantom{\rule{0.2em}{0ex}}\theta -\text{sin}\phantom{\rule{0.2em}{0ex}}\theta}$
${y}^{2}=4x$
For the following exercises, convert the polar equation to rectangular form and sketch its graph.
$r=4\phantom{\rule{0.2em}{0ex}}\text{sin}\phantom{\rule{0.2em}{0ex}}\theta $
${x}^{2}+{y}^{2}=4y$
$r=6\phantom{\rule{0.2em}{0ex}}\text{cos}\phantom{\rule{0.2em}{0ex}}\theta $
$r=\theta $
$x\phantom{\rule{0.2em}{0ex}}\text{tan}\sqrt{{x}^{2}+{y}^{2}}=y$
$r=\text{cot}\phantom{\rule{0.2em}{0ex}}\theta \phantom{\rule{0.2em}{0ex}}\text{csc}\phantom{\rule{0.2em}{0ex}}\theta $
For the following exercises, sketch a graph of the polar equation and identify any symmetry.
$r=1+\text{sin}\phantom{\rule{0.2em}{0ex}}\theta $
y -axis symmetry
$r=3-2\phantom{\rule{0.2em}{0ex}}\text{cos}\phantom{\rule{0.2em}{0ex}}\theta $
$r=2-2\phantom{\rule{0.2em}{0ex}}\text{sin}\phantom{\rule{0.2em}{0ex}}\theta $
y -axis symmetry
$r=5-4\phantom{\rule{0.2em}{0ex}}\text{sin}\phantom{\rule{0.2em}{0ex}}\theta $
$r=3\phantom{\rule{0.2em}{0ex}}\text{cos}\left(2\theta \right)$
x - and
y -axis symmetry and symmetry about the pole
$r=3\phantom{\rule{0.2em}{0ex}}\text{sin}\left(2\theta \right)$
$r=2\phantom{\rule{0.2em}{0ex}}\text{cos}\left(3\theta \right)$
x -axis symmetry
$r=3\phantom{\rule{0.2em}{0ex}}\text{cos}\left(\frac{\theta}{2}\right)$
${r}^{2}=4\phantom{\rule{0.2em}{0ex}}\text{cos}\left(2\theta \right)$
x - and
y -axis symmetry and symmetry about the pole
${r}^{2}=4\phantom{\rule{0.2em}{0ex}}\text{sin}\phantom{\rule{0.2em}{0ex}}\theta $
[T] The graph of $r=2\phantom{\rule{0.2em}{0ex}}\text{cos}(2\theta )\text{sec}(\theta ).$ is called a strophoid. Use a graphing utility to sketch the graph, and, from the graph, determine the asymptote.
[T] Use a graphing utility and sketch the graph of $r=\frac{6}{2\phantom{\rule{0.2em}{0ex}}\text{sin}\phantom{\rule{0.2em}{0ex}}\theta -3\phantom{\rule{0.2em}{0ex}}\text{cos}\phantom{\rule{0.2em}{0ex}}\theta}.$
a line
[T] Use a graphing utility to graph $r=\frac{1}{1-\text{cos}\phantom{\rule{0.2em}{0ex}}\theta}.$
[T] Use technology to graph $r={e}^{\text{sin}(\theta )}-2\phantom{\rule{0.2em}{0ex}}\text{cos}\left(4\theta \right).$
[T] Use technology to plot $r=\text{sin}\left(\frac{3\theta}{7}\right)$ (use the interval $0\le \theta \le 14\pi ).$
Without using technology, sketch the polar curve $\theta =\frac{2\pi}{3}.$
[T] Use a graphing utility to plot $r=\theta \phantom{\rule{0.2em}{0ex}}\text{sin}\phantom{\rule{0.2em}{0ex}}\theta $ for $\text{\u2212}\pi \le \theta \le \pi .$
[T] Use technology to plot $r={e}^{\mathrm{-0.1}\theta}$ for $\mathrm{-10}\le \theta \le 10.$
[T] There is a curve known as the “ Black Hole .” Use technology to plot $r={e}^{\mathrm{-0.01}\theta}$ for $\mathrm{-100}\le \theta \le 100.$
[T] Use the results of the preceding two problems to explore the graphs of $r={e}^{\mathrm{-0.001}\theta}$ and $r={e}^{\mathrm{-0.0001}\theta}$ for $\left|\theta \right|>100.$
Answers vary. One possibility is the spiral lines become closer together and the total number of spirals increases.
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