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Q. 16

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Found in: Page 583

### Precalculus Enhanced with Graphing Utilities

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

# In Problems 13–28, transform each polar equation to an equation in rectangular coordinates. Then identify and graph the equation. Verify your graph using a graphing utility. $\theta =-\frac{\mathrm{\pi }}{4}$

The equation in regular coordinates is $y=-x.$

The equation is of a line passing through the pole making an angle of $-\frac{\mathrm{\pi }}{4}$ with polar axis and the graph of the equation is

The graph is the same as the graph we get in graphing utility.

See the step by step solution

## Step 1. Given Information

The given polar equation is $\theta =-\frac{\mathrm{\pi }}{4}.$

We have to transform each polar equation to an equation in rectangular coordinates then identify and graph the equation then verified the graph by using a graphing utility.

## Step 2. Transforming polar equation to an equation in rectangular coordinates

To transform the equation to rectangular coordinates, take the tangent on both sides

$\mathrm{tan}\theta =\mathrm{tan}\left(-\frac{\mathrm{\pi }}{4}\right)\phantom{\rule{0ex}{0ex}}\frac{y}{x}=-1\left[\mathrm{tan}\theta =\frac{y}{x},\mathrm{tan}\left(-\frac{\mathrm{\pi }}{4}\right)=-1\right]\phantom{\rule{0ex}{0ex}}y=-x$

## Step 3. Identifying and graphing the equation

The graph of $\theta =-\frac{\mathrm{\pi }}{4}$ is a line passing through the pole making an angle of $-\frac{\mathrm{\pi }}{4}$ with the polar axis.

The graph of the equation is

## Step 4. Verifying the graph by graphing utility

The graph of the equation by the graphing utility is

Hence it is verified that the graph is the same.