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Found in: Page 522

Essential Calculus: Early Transcendentals

Book edition 2nd
Author(s) James Stewart
Pages 830 pages
ISBN 9781133112280

Plot the point whose polar coordinates are given. Then find two other pairs of polar coordinates of this point, one with $$r > 0$$and one with $$r < 0$$.\begin{aligned}{l}(a)(1,7\pi /4)\\(b)( - 3,\pi /6)\\(c)(1, - 1)\end{aligned}

1. The other two coordinates are$$(1, - \frac{\pi }{4}),( - 1,\frac{{3\pi }}{4})$$
2. The other two coordinates are$$( - 3,\frac{{13\pi }}{6}),(3,\frac{{7\pi }}{6})$$
3. The other two coordinates are$$(1,2\pi - 1),( - 1,\pi - 1)$$
See the step by step solution

Step 1: Plotting the points

1. The polar plot for the point$$(1,\frac{{7\pi }}{4})$$ is shown below:

The coordinates $$(r,\theta + 2n\pi ){\rm{ and }}(r,\theta )$$represent the same point.

Another coordinates for this point are$$\left( {1,\frac{{7\pi }}{4} - 2\pi } \right) = \left( {1, - \frac{\pi }{4}} \right)$$

The coordinates $$( - r,\theta + (2n + 1)\pi ){\rm{ and }}(r,\theta )$$represent the same point.

Another coordinates for this point are $$\left( { - 1,\frac{{7\pi }}{4} - \pi } \right) = \left( { - 1,\frac{{3\pi }}{4}} \right)$$

Step 2: Plotting the points

b. The polar plot for the point$$( - 3,\frac{\pi }{6})$$is shown below:

The coordinates $$(r,\theta + 2n\pi ){\rm{ and }}(r,\theta )$$represent the same point.

Another coordinates for this point are$$\left( { - 3,\frac{\pi }{6} + 2\pi } \right) = \left( { - 3,\frac{{13\pi }}{6}} \right)$$

The coordinates $$( - r,\theta + (2n + 1)\pi ){\rm{ and }}(r,\theta )$$represent the same point.

Another coordinates for this point are $$\left( {3,\frac{\pi }{6} + \pi } \right) = \left( {3,\frac{{7\pi }}{6}} \right)$$

Step 2: Plotting the points

(c)The polar plot for the point$$(1, - 1)$$ is shown below:

The coordinates $$(r,\theta + 2n\pi ){\rm{ and }}(r,\theta )$$represent the same point.

Another coordinates for this point are$$(1,2\pi - 1)$$

The coordinates $$( - r,\theta + (2n + 1)\pi ){\rm{ and }}(r,\theta )$$represent the same point.

Another coordinates for this point are $$( - 1,\pi - 1)$$