Problem Set: Polar Coordinates

In the following exercises, plot the point whose polar coordinates are given by first constructing the angle $\theta$ and then marking off the distance r along the ray.

1. $\left(3,\frac{\pi }{6}\right)$

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$\left(-2,\frac{5\pi }{3}\right)$

3. $\left(0,\frac{7\pi }{6}\right)$

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$\left(-4,\frac{3\pi }{4}\right)$

5. $\left(1,\frac{\pi }{4}\right)$

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$\left(2,\frac{5\pi }{6}\right)$

7. $\left(1,\frac{\pi }{2}\right)$

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For the following exercises, consider the polar graph below. Give two sets of polar coordinates for each point.

The polar coordinate plane is divided into 12 pies. Point A is drawn on the first circle on the first spoke above the θ = 0 line in the first quadrant. Point B is drawn in the fourth quadrant on the third circle and the second spoke below the θ = 0 line. Point C is drawn on the θ = π line on the third circle. Point D is drawn on the fourth circle on the first spoke below the θ = π line.

8. Coordinates of point A.

9. Coordinates of point B.

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$B\begin{array}{cc}\left(3,\frac{\text{-}\pi }{3}\right)\hfill & B\left(-3,\frac{2\pi }{3}\right)\hfill \end{array}$

10. Coordinates of point C.

11. Coordinates of point D.

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$D\left(5,\frac{7\pi }{6}\right)D\left(-5,\frac{\pi }{6}\right)$

For the following exercises, the rectangular coordinates of a point are given. Find two sets of polar coordinates for the point in $\left(0,2\pi \right]$ . Round to three decimal places.

12.

$\left(2,2\right)$

13. $\left(3,-4\right)$

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$\begin{array}{cc}\left(5,-0.927\right)\hfill & \left(-5,-0.927+\pi \right)\hfill \end{array}$

14.

$\left(8,15\right)$

15. $\left(-6,8\right)$

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$\left(10,-0.927\right)\left(-10,-0.927+\pi \right)$

16.

$\left(4,3\right)$

17. $\left(3,\text{-}\sqrt{3}\right)$

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$\left(2\sqrt{3},-0.524\right)\left(-2\sqrt{3},-0.524+\pi \right)$

For the following exercises, find rectangular coordinates for the given point in polar coordinates.

18.

$\left(2,\frac{5\pi }{4}\right)$

19. $\left(-2,\frac{\pi }{6}\right)$

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$\left(\begin{array}{cc}\text{-}\sqrt{3},\hfill & -1\hfill \end{array}\right)$

20.

$\left(5,\frac{\pi }{3}\right)$

21. $\left(1,\frac{7\pi }{6}\right)$

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$\left(\begin{array}{cc}-\frac{\sqrt{3}}{2},\hfill & \frac{-1}{2}\hfill \end{array}\right)$

22.

$\left(-3,\frac{3\pi }{4}\right)$

23. $\left(0,\frac{\pi }{2}\right)$

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$\left(\begin{array}{cc}0,\hfill & 0\hfill \end{array}\right)$

24.

$\left(-4.5,6.5\right)$

For the following exercises, determine whether the graphs of the polar equation are symmetric with respect to the $x$ -axis, the $y$ -axis, or the origin.

25. $r=3\sin\left(2\theta \right)$

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26.

${r}^{2}=9\cos\theta$

27. $r=\cos\left(\frac{\theta }{5}\right)$

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28.

$r=2\sec\theta$

29. $r=1+\cos\theta$

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For the following exercises, describe the graph of each polar equation. Confirm each description by converting into a rectangular equation.

30.

$r=3$

31. $\theta =\frac{\pi }{4}$

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Line

$y=x$

32.

$r=\sec\theta$

33. $r=\csc\theta$

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$y=1$

For the following exercises, convert the rectangular equation to polar form and sketch its graph.

34.

${x}^{2}+{y}^{2}=16$

35. ${x}^{2}-{y}^{2}=16$

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Hyperbola; polar form

${r}^{2}\cos\left(2\theta \right)=16$ or

${r}^{2}=16\sec\theta$ .

A hyperbola with vertices at (−4, 0) and (4, 0), the first pointing out into quadrants II and III and the second pointing out into quadrants I and IV.

36.

$x=8$

For the following exercises, convert the rectangular equation to polar form and sketch its graph.

37. $3x-y=2$

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$r=\frac{2}{3\cos\theta -\sin\theta }$

A straight line with slope 3 and y intercept −2.

38.

${y}^{2}=4x$

For the following exercises, convert the polar equation to rectangular form and sketch its graph.

39. $r=4\sin\theta$

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${x}^{2}+{y}^{2}=4y$

A circle of radius 2 with center at (2, π/2).

40. $r=6\cos\theta$

41. $r=\theta$

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$x\tan\sqrt{{x}^{2}+{y}^{2}}=y$

A spiral starting at the origin and crossing θ = π/2 between 1 and 2, θ = π between 3 and 4, θ = 3π/2 between 4 and 5, θ = 0 between 6 and 7, θ = π/2 between 7 and 8, and θ = π between 9 and 10.

42. $r=\cot\theta \csc\theta$

For the following exercises, sketch a graph of the polar equation and identify any symmetry.

43. $r=1+\sin\theta$

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44.

$r=3 - 2\cos\theta$

45. $r=2 - 2\sin\theta$

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46.

$r=5 - 4\sin\theta$

47. $r=3\cos\left(2\theta \right)$

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48.

$r=3\sin\left(2\theta \right)$

49. $r=2\cos\left(3\theta \right)$

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50.

$r=3\cos\left(\frac{\theta }{2}\right)$

51. ${r}^{2}=4\cos\left(2\theta \right)$

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52.

${r}^{2}=4\sin\theta$

53. $r=2\theta$

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54. [T] The graph of

$r=2\cos\left(2\theta \right)\sec\left(\theta \right)$ . is called a strophoid. Use a graphing utility to sketch the graph, and, from the graph, determine the asymptote.

55. [T] Use a graphing utility and sketch the graph of $r=\frac{6}{2\sin\theta -3\cos\theta }$ .

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56. [T] Use a graphing utility to graph

$r=\frac{1}{1-\cos\theta }$ .

57. [T] Use technology to graph $r={e}^{\sin\left(\theta \right)}-2\cos\left(4\theta \right)$ .

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58. [T] Use technology to plot

$r=\sin\left(\frac{3\theta }{7}\right)$ (use the interval

$0\le \theta \le 14\pi$ ).

59. Without using technology, sketch the polar curve $\theta =\frac{2\pi }{3}$ .

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60. [T] Use a graphing utility to plot

$r=\theta \sin\theta$ for

$\text{-}\pi \le \theta \le \pi$ .

61. [T] Use technology to plot $r={e}^{-0.1\theta }$ for $-10\le \theta \le 10$ .

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62. [T] There is a curve known as the “Black Hole.” Use technology to plot

$r={e}^{-0.01\theta }$ for

$-100\le \theta \le 100$ .

63. [T] Use the results of the preceding two problems to explore the graphs of $r={e}^{-0.001\theta }$ and $r={e}^{-0.0001\theta }$ for $|\theta |>100$ .

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Parametric Equations and Polar Coordinates: Supplemental Content

Problem Set: Polar Coordinates

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