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Q. 64

Expert-verified
Found in: Page 584

### Precalculus Enhanced with Graphing Utilities

Book edition 6th
Author(s) Sullivan
Pages 1200 pages
ISBN 9780321795465

# In Problems 61–66, graph each pair of polar equations on the same polar grid. Find the polar coordinates of the point(s) of intersection and label the point(s) on the graph.$r=3;\phantom{\rule{0ex}{0ex}}r=2+2\mathrm{cos}\theta$

The graph of the polar equations on the same polar grid is:

See the step by step solution

## Step 1. Given

The polar equations:

$r=3;\phantom{\rule{0ex}{0ex}}r=2+2\mathrm{cos}\theta$

## Step 2. Substitute the values and equate them.

Substitute the values of $r$ and equate them.

$3=2+2\mathrm{cos}\theta \phantom{\rule{0ex}{0ex}}2\mathrm{cos}\theta =1\phantom{\rule{0ex}{0ex}}\mathrm{cos}\theta =\frac{1}{2}\phantom{\rule{0ex}{0ex}}\theta ={\mathrm{cos}}^{-1}\left(\frac{1}{2}\right)\phantom{\rule{0ex}{0ex}}=\frac{\mathrm{\pi }}{3},\frac{5\mathrm{\pi }}{3}$

The point of intersection is $\left(3,\frac{\mathrm{\pi }}{3}\right),\left(3,\frac{5\mathrm{\pi }}{3}\right)$

## Step 3. Sketch the graph of r=3

Sketch the graph of $r=3$

## Step 4. Polar axis

Replace $\theta by-\theta$

$r=2+2\mathrm{cos}\left(-\theta \right)\phantom{\rule{0ex}{0ex}}=2+2\mathrm{cos}\theta$

The test is satisfied, so the graph is symmetric with respect to the polar axis.

## Step 5. The line

Replace $\theta$ by $\mathrm{\pi }-\mathrm{\theta }$

$r=2+2\mathrm{cos}\left(\pi -\theta \right)\phantom{\rule{0ex}{0ex}}=2+2\left(\mathrm{cos}\pi .\mathrm{cos}\theta +\mathrm{sin}\pi .\mathrm{sin}\theta \right)\phantom{\rule{0ex}{0ex}}=2+2\left(\left(-1\right)\mathrm{cos}\theta +0\right)\phantom{\rule{0ex}{0ex}}=2-2\mathrm{cos}\theta$

The test fails, so the graph may or may not be symmetric with respect to the line $\frac{\mathrm{\pi }}{2}$

## Step 6. The pole

Replace $rby-r$

$-r=2+2\mathrm{cos}\theta \phantom{\rule{0ex}{0ex}}r=-2-2\mathrm{cos}\theta$

The test fails, so the graph may or may not be symmetric with respect to the pole.

## Step 7. Sketch the graph r=2+2 cos θ

Graph the polar function $r=2+2\mathrm{cos}\theta$ on the same polar grid.