# 11.3E: Exercises for Section 11.3

• Gilbert Strang & Edwin “Jed” Herman
• OpenStax

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In exercises 1 - 7, 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) $$\left(3,\frac{π}{6}\right)$$

2) $$\left(−2,\frac{5π}{3}\right)$$

3) $$\left(0,\frac{7π}{6}\right)$$

4) $$\left(−4,\frac{3π}{4}\right)$$

5) $$\left(1,\frac{π}{4}\right)$$

6) $$\left(2,\frac{5π}{6}\right)$$

7) $$\left(1,\frac{π}{2}\right)$$

In exercises 8 - 11, consider the polar graph below. Give two sets of polar coordinates for each point.

8) Coordinates of point A.

9) Coordinates of point B.

$$B\left(3,\frac{−π}{3}\right) B\left(−3,\frac{2π}{3}\right)$$

10) Coordinates of point C.

11) Coordinates of point D.

$$D\left(5,\frac{7π}{6}\right) D\left(−5,\frac{π}{6}\right)$$

In exercises 12 - 17, the rectangular coordinates of a point are given. Find two sets of polar coordinates for the point in $$(0,2π]$$. Round to three decimal places.

12) $$(2,2)$$

13) $$(3,−4)$$

$$(5,−0.927),\;(−5,−0.927+π)$$

14) $$(8,15)$$

15) $$(−6,8)$$

$$(10,−0.927),\;(−10,−0.927+π)$$

16) $$(4,3)$$

17) $$(3,−\sqrt{3})$$

$$(2\sqrt{3},−0.524),\;(−2\sqrt{3},−0.524+π)$$

In exercises 18 - 24, find rectangular coordinates for the given point in polar coordinates.

18) $$\left(2,\frac{5π}{4}\right)$$

19) $$\left(−2,\frac{π}{6}\right)$$

$$(−\sqrt{3},−1)$$

20) $$\left(5,\frac{π}{3}\right)$$

21) $$\left(1,\frac{7π}{6}\right)$$

$$\left(−\frac{\sqrt{3}}{2},\frac{−1}{2}\right)$$

22) $$\left(−3,\frac{3π}{4}\right)$$

23) $$\left(0,\frac{π}{2}\right)$$

$$(0,0)$$

24) $$(−4.5,6.5)$$

In exercises 25 - 29, determine whether the graphs of the polar equation are symmetric with respect to the $$x$$-axis, the $$y$$-axis, or the origin.

25) $$r=3\sin(2θ)$$

Symmetry with respect to the x-axis, y-axis, and origin.

26) $$r^2=9\cos θ$$

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

Symmetric with respect to x-axis only.

28) $$r=2\sec θ$$

29) $$r=1+\cos θ$$

Symmetry with respect to x-axis only.

In exercises 30 - 33, describe the graph of each polar equation. Confirm each description by converting into a rectangular equation.

30) $$r=3$$

31) $$θ=\frac{π}{4}$$

Line $$y=x$$

32) $$r=\sec θ$$

33) $$r=\csc θ$$

$$y=1$$

In exercises 34 - 36, convert the rectangular equation to polar form and sketch its graph.

34) $$x^2+y^2=16$$

35) $$x^2−y^2=16$$

Hyperbola; polar form $$r^2\cos(2θ)=16$$ or $$r^2=16\sec θ.$$

36) $$x=8$$

In exercises 37 - 38, convert the rectangular equation to polar form and sketch its graph.

37) $$3x−y=2$$

$$r=\frac{2}{3\cos θ−\sin θ}$$

38) $$y^2=4x$$

In exercises 39 - 43, convert the polar equation to rectangular form and sketch its graph.

39) $$r=4\sin θ$$

40) $$x^2+y^2=4y$$

41) $$r=6\cos θ$$

42) $$r=θ$$

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

43) $$r=\cot θ\csc θ$$

In exercises 44 - 54, sketch a graph of the polar equation and identify any symmetry.

44) $$r=1+\sin θ$$

$$y$$-axis symmetry

45) $$r=3−2\cos θ$$

46) $$r=2−2\sin θ$$

$$y$$-axis symmetry

47) $$r=5−4\sin θ$$

48) $$r=3\cos(2θ)$$

$$x$$-and $$y$$-axis symmetry and symmetry about the pole

49) $$r=3\sin(2θ)$$

50) $$r=2\cos(3θ)$$

$$x$$-axis symmetry

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

52) $$r^2=4\cos\left(\frac{2}{θ}\right)$$

$$x$$-and $$y$$-axis symmetry and symmetry about the pole

53) $$r^2=4\sin θ$$

54) $$r=2θ$$

no symmetry

55) [T] The graph of $$r=2\cos(2θ)\sec(θ).$$ is called a strophoid. Use a graphing utility to sketch the graph, and, from the graph, determine the asymptote.

56) [T] Use a graphing utility and sketch the graph of $$r=\dfrac{6}{2\sin θ−3\cos θ}$$.

a line

57) [T] Use a graphing utility to graph $$r=\frac{1}{1−\cos θ}$$.

58) [T] Use technology to graph $$r=e^{\sin(θ)}−2\cos(4θ)$$.

59) [T] Use technology to plot $$r=\sin(\frac{3θ}{7})$$ (use the interval $$0≤θ≤14π$$).

60) Without using technology, sketch the polar curve $$θ=\frac{2π}{3}$$.

61) [T] Use a graphing utility to plot $$r=θ\sin θ$$ for $$−π≤θ≤π$$.

62) [T] Use technology to plot $$r=e^{−0.1θ}$$ for $$−10≤θ≤10.$$

63) [T] There is a curve known as the “Black Hole.” Use technology to plot $$r=e^{−0.01θ}$$ for $$−100≤θ≤100$$.
64) [T] Use the results of the preceding two problems to explore the graphs of $$r=e^{−0.001θ}$$ and $$r=e^{−0.0001θ}$$ for $$|θ|>100$$.