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How are the polar axes different from the x - and y -axes of the Cartesian plane?
Explain how polar coordinates are graphed.
Determine $\text{\hspace{0.17em}}\theta \text{\hspace{0.17em}}$ for the point, then move $\text{\hspace{0.17em}}r\text{\hspace{0.17em}}$ units from the pole to plot the point. If $\text{\hspace{0.17em}}r\text{\hspace{0.17em}}$ is negative, move $\text{\hspace{0.17em}}r\text{\hspace{0.17em}}$ units from the pole in the opposite direction but along the same angle. The point is a distance of $\text{\hspace{0.17em}}r\text{\hspace{0.17em}}$ away from the origin at an angle of $\text{\hspace{0.17em}}\theta \text{\hspace{0.17em}}$ from the polar axis.
How are the points $\text{\hspace{0.17em}}\left(3,\frac{\pi}{2}\right)\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}\left(-3,\frac{\pi}{2}\right)\text{\hspace{0.17em}}$ related?
Explain why the points $\text{\hspace{0.17em}}\left(-3,\frac{\pi}{2}\right)\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}\left(3,-\frac{\pi}{2}\right)\text{\hspace{0.17em}}$ are the same.
The point $\text{\hspace{0.17em}}\left(-3,\frac{\pi}{2}\right)\text{\hspace{0.17em}}$ has a positive angle but a negative radius and is plotted by moving to an angle of $\text{\hspace{0.17em}}\frac{\pi}{2}\text{\hspace{0.17em}}$ and then moving 3 units in the negative direction. This places the point 3 units down the negative y -axis. The point $\text{\hspace{0.17em}}\left(3,-\frac{\pi}{2}\right)\text{\hspace{0.17em}}$ has a negative angle and a positive radius and is plotted by first moving to an angle of $\text{\hspace{0.17em}}-\frac{\pi}{2}\text{\hspace{0.17em}}$ and then moving 3 units down, which is the positive direction for a negative angle. The point is also 3 units down the negative y -axis.
For the following exercises, convert the given polar coordinates to Cartesian coordinates with $\text{\hspace{0.17em}}r>0\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}0\le \theta \le 2\pi .\text{\hspace{0.17em}}$ Remember to consider the quadrant in which the given point is located when determining $\text{\hspace{0.17em}}\theta \text{\hspace{0.17em}}$ for the point.
$\left(7,\frac{7\pi}{6}\right)$
$\left(6,-\frac{\pi}{4}\right)$
$\left(-3,\frac{\pi}{6}\right)$
$\left(-\frac{3\sqrt{3}}{2},-\frac{3}{2}\right)$
$\left(4,\frac{7\pi}{4}\right)$
For the following exercises, convert the given Cartesian coordinates to polar coordinates with $\text{\hspace{0.17em}}r>0,\text{\hspace{0.17em}}\text{\hspace{0.17em}}0\le \theta <2\pi .\text{\hspace{0.17em}}$ Remember to consider the quadrant in which the given point is located.
$\left(-4,6\right)$
$\left(\mathrm{-10},\mathrm{-13}\right)$
For the following exercises, convert the given Cartesian equation to a polar equation.
$y=4{x}^{2}$
$y=2{x}^{4}$
$r=\sqrt[3]{\frac{sin\theta}{2co{s}^{4}\theta}}$
${x}^{2}+{y}^{2}=4y$
${x}^{2}-{y}^{2}=x$
${x}^{2}-{y}^{2}=3y$
$r=\frac{3\mathrm{sin}\theta}{\mathrm{cos}\left(2\theta \right)}$
${x}^{2}+{y}^{2}=9$
${x}^{2}=9y$
$r=\frac{9\mathrm{sin}\theta}{{\mathrm{cos}}^{2}\theta}$
${y}^{2}=9x$
$9xy=1$
$r=\sqrt{\frac{1}{9\mathrm{cos}\theta \mathrm{sin}\theta}}$
For the following exercises, convert the given polar equation to a Cartesian equation. Write in the standard form of a conic if possible, and identify the conic section represented.
$r=3\mathrm{sin}\text{\hspace{0.17em}}\theta $
$r=4\mathrm{cos}\text{\hspace{0.17em}}\theta $
${x}^{2}+{y}^{2}=4x\text{\hspace{0.17em}}$ or $\text{\hspace{0.17em}}\frac{{\left(x-2\right)}^{2}}{4}+\frac{{y}^{2}}{4}=1;$ circle
$r=\frac{4}{\mathrm{sin}\text{\hspace{0.17em}}\theta +7\mathrm{cos}\text{\hspace{0.17em}}\theta}$
$r=\frac{6}{\mathrm{cos}\text{\hspace{0.17em}}\theta +3\mathrm{sin}\text{\hspace{0.17em}}\theta}$
$3y+x=6;\text{\hspace{0.17em}}$ line
$r=2\mathrm{sec}\text{\hspace{0.17em}}\theta $
$r=3\mathrm{csc}\text{\hspace{0.17em}}\theta $
$y=3;\text{\hspace{0.17em}}$ line
$r=\sqrt{r\mathrm{cos}\text{\hspace{0.17em}}\theta +2}$
${r}^{2}=4\mathrm{sec}\text{\hspace{0.17em}}\theta \text{\hspace{0.17em}}\mathrm{csc}\text{\hspace{0.17em}}\theta $
$xy=4;\text{\hspace{0.17em}}$ hyperbola
${r}^{2}=4$
${x}^{2}+{y}^{2}=4;\text{\hspace{0.17em}}$ circle
$r=\frac{1}{4\mathrm{cos}\text{\hspace{0.17em}}\theta -3\mathrm{sin}\text{\hspace{0.17em}}\theta}$
$r=\frac{3}{\mathrm{cos}\text{\hspace{0.17em}}\theta -5\mathrm{sin}\text{\hspace{0.17em}}\theta}$
$x-5y=3;\text{\hspace{0.17em}}$ line
For the following exercises, find the polar coordinates of the point.
For the following exercises, plot the points.
$\left(-1,-\frac{\pi}{2}\right)$
$\left(-4,\frac{\pi}{3}\right)$
$\left(4,\frac{-5\pi}{4}\right)$
$\left(-1.5,\frac{7\pi}{6}\right)$
$\left(1,\frac{3\pi}{2}\right)$
For the following exercises, convert the equation from rectangular to polar form and graph on the polar axis.
$5x-y=6$
$r=\frac{6}{5\mathrm{cos}\theta -\mathrm{sin}\theta}$
$2x+7y=-3$
${\left(x+2\right)}^{2}+{\left(y+3\right)}^{2}=13$
${x}^{2}+{y}^{2}=5y$
For the following exercises, convert the equation from polar to rectangular form and graph on the rectangular plane.
$\theta =-\frac{2\pi}{3}$
$r=\mathrm{sec}\text{\hspace{0.17em}}\theta $
$r=\mathrm{-10}\mathrm{sin}\text{\hspace{0.17em}}\theta $
${x}^{2}+{\left(y+5\right)}^{2}=25$
$r=3\mathrm{cos}\text{\hspace{0.17em}}\theta $
Use a graphing calculator to find the rectangular coordinates of $\text{\hspace{0.17em}}\left(2,-\frac{\pi}{5}\right).\text{\hspace{0.17em}}$ Round to the nearest thousandth.
$\left(1.618,-1.176\right)$
Use a graphing calculator to find the rectangular coordinates of $\text{\hspace{0.17em}}\left(-3,\frac{3\pi}{7}\right).\text{\hspace{0.17em}}$ Round to the nearest thousandth.
Use a graphing calculator to find the polar coordinates of $\text{\hspace{0.17em}}\left(-7,8\right)\text{\hspace{0.17em}}$ in degrees. Round to the nearest thousandth.
$\left(10.630,\mathrm{131.186\xb0}\right)$
Use a graphing calculator to find the polar coordinates of $\text{\hspace{0.17em}}\left(3,-4\right)\text{\hspace{0.17em}}$ in degrees. Round to the nearest hundredth.
Use a graphing calculator to find the polar coordinates of $\text{\hspace{0.17em}}\left(-2,0\right)\text{\hspace{0.17em}}$ in radians. Round to the nearest hundredth.
$\text{\hspace{0.17em}}\left(2,3.14\right)or\left(2,\pi \right)\text{\hspace{0.17em}}$
Describe the graph of $\text{\hspace{0.17em}}r=a\mathrm{sec}\text{\hspace{0.17em}}\theta ;a>0.$
Describe the graph of $\text{\hspace{0.17em}}r=a\mathrm{sec}\text{\hspace{0.17em}}\theta ;a<0.$
A vertical line with $\text{\hspace{0.17em}}a\text{\hspace{0.17em}}$ units left of the y -axis.
Describe the graph of $\text{\hspace{0.17em}}r=a\mathrm{csc}\text{\hspace{0.17em}}\theta ;a>0.$
Describe the graph of $\text{\hspace{0.17em}}r=a\mathrm{csc}\text{\hspace{0.17em}}\theta ;a<0.$
A horizontal line with $\text{\hspace{0.17em}}a\text{\hspace{0.17em}}$ units below the x -axis.
What polar equations will give an oblique line?
For the following exercise, graph the polar inequality.
$0\le \theta \le \frac{\pi}{4}$
$\theta =\frac{\pi}{4},\text{\hspace{0.17em}}r\text{\hspace{0.17em}}\ge \text{\hspace{0.17em}}2$
$\theta =\frac{\pi}{4},\text{\hspace{0.17em}}r\text{\hspace{0.17em}}\ge \mathrm{-3}$
$0\le \theta \le \frac{\pi}{3},\text{\hspace{0.17em}}r\text{\hspace{0.17em}}<\text{\hspace{0.17em}}2$
$\frac{-\pi}{6}<\theta \le \frac{\pi}{3},-3<r\text{\hspace{0.17em}}<\text{\hspace{0.17em}}2$
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