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Q. 16

Expert-verifiedFound in: Page 583

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.

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.

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

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

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

The graph of the equation by the graphing utility is

Hence it is verified that the graph is the same.

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