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Q. 16

Expert-verified
Found in: Page 583

Precalculus Enhanced with Graphing Utilities

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

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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.

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