4.3E:

Polar Coordinates

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

1) \(\displaystyle (3,\frac{π}{6})\)

Solution:

2) \(\displaystyle (−2,\frac{5π}{3})\)

3) \(\displaystyle (0,\frac{7π}{6})\)

Solution:

4) \(\displaystyle (−4,\frac{3π}{4})\)

5) \(\displaystyle (1,\frac{π}{4})\)

Solution:

6) \(\displaystyle (2,\frac{5π}{6})\)

7) \(\displaystyle (1,\frac{π}{2})\)

Solution:

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

8) Coordinates of point A.

9) Coordinates of point B.

Solution: \(\displaystyle B(3,\frac{−π}{3}) B(−3,\frac{2π}{3})\)

10) Coordinates of point C.

11) Coordinates of point D.

Solution: \(\displaystyle D(5,\frac{7π}{6}) D(−5,\frac{π}{6})\)

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

12) \(\displaystyle (2,2)\)

13) \(\displaystyle (3,−4) (3, −4)\)

Solution: \(\displaystyle (5,−0.927)(−5,−0.927+π)\)

14) \(\displaystyle (8,15)\)

15) \(\displaystyle (−6,8)\)

Solution: \(\displaystyle (10,−0.927)(−10,−0.927+π)\)

16) \(\displaystyle (4,3)\)

17) \(\displaystyle (3,−\sqrt{3})\)

Solution: \(\displaystyle 2\sqrt{3},−0.524)(−2\sqrt{3},−0.524+π)\)

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

18) \(\displaystyle (2,\frac{5π}{4})\)

19) \(\displaystyle (−2,\frac{π}{6})\)

Solution: \(\displaystyle −\sqrt{3},−1)\)

20) \(\displaystyle (5,\frac{π}{3})\)

21) \(\displaystyle (1,\frac{7π}{6})\)

Solution: \(\displaystyle (−\frac{\sqrt{3}}{2},\frac{−1}{2})\)

22) \(\displaystyle (−3,\frac{3π}{4})\)

23) \(\displaystyle (0,\frac{π}{2})\)

Solution: \(\displaystyle (0,0)\)

24) \(\displaystyle (−4.5,6.5)\)

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

25) \(\displaystyle r=3sin(2θ)\)

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

26) \(\displaystyle r^2=9cosθ\)

27) \(\displaystyle r=cos(\frac{θ}{5})\)

Solution: Symmetric with respect to x-axis only.

28) \(\displaystyle r=2secθ\)

29) \(\displaystyle r=1+cosθ\)

Solution: Symmetry with respect to x-axis only.

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

30) \(\displaystyle r=3\)

31) \(\displaystyle θ=\frac{π}{4}\)

Solution: Line \(\displaystyle y=x\)

32) \(\displaystyle r=secθ\)

33) \(\displaystyle r=cscθ\)

Solution: \(\displaystyle y=1\)

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

34) \(\displaystyle x^2+y^2=16\)

35) \(\displaystyle x^2−y^2=16\)

Solution: Hyperbola; polar form \(\displaystyle r^2cos(2θ)=16\) or \(\displaystyle r^2=16secθ.\)

36) \(\displaystyle x=8\)

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

37) \(\displaystyle 3x−y=2\)

Solution: \(\displaystyle r=\frac{2}{3cosθ−sinθ}\)

38) \(\displaystyle y^2=4x\)

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

39) \(\displaystyle r=4sinθ\)

40) \(\displaystyle x^2+y^2=4y\)

Solution:

41) \(\displaystyle r=6cosθ\)

42) \(\displaystyle r=θ\)

Solution: \(\displaystyle xtan\sqrt{x^2+y^2}=y\)

43) \(\displaystyle r=cotθcscθ\)

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

44) \(\displaystyle r=1+sinθ\)

Solution: y-axis symmetry

45) \(\displaystyle r=3−2cosθ\)

46) \(\displaystyle r=2−2sinθ\)

Solution: y-axis symmetry

47) \(\displaystyle r=5−4sinθ\)

48) \(\displaystyle r=3cos(2θ)\)

Solution: x- and y-axis symmetry and symmetry about the pole

49) \(\displaystyle r=3sin(2θ)\)

50) \(\displaystyle r=2cos(3θ)\)

Solution: x-axis symmetry

51) \(\displaystyle r=3cos(\frac{θ}{2})\)

52) \(\displaystyle r^2=4cos(\frac{2}{θ})\)

Solution: x- and y-axis symmetry and symmetry about the pole

53) \(\displaystyle r^2=4sinθ\)

54) \(\displaystyle r=2θ\)

Solution: no symmetry

55) [T] The graph of \(\displaystyle r=2cos(2θ)sec(θ).\) 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 \(\displaystyle r=\frac{6}{2sinθ−3cosθ}\).

Solution: a line

57) [T] Use a graphing utility to graph \(\displaystyle r=\frac{1}{1−cosθ}\).

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

Solution:

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

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

Solution:

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

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

Solution:

63) [T] There is a curve known as the “Black Hole.” Use technology to plot \(\displaystyle r=e^{−0.01θ}\) for \(\displaystyle −100≤θ≤100\).

64) [T] Use the results of the preceding two problems to explore the graphs of \(\displaystyle r=e^{−0.001θ}\) and \(\displaystyle r=e^{−0.0001θ}\) for \(\displaystyle |θ|>100\).

Solution: Answers vary. One possibility is the spiral lines become closer together and the total number of spirals increases.