5.3E: Exercises

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

Answer: $On the polar coordinate plane, a ray is drawn from the origin marking π/6 and a point is drawn when this line crosses the circle with radius 3.$

2) $\left(−2,\frac{5 \pi }{3}\right)$

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

Answer: $On the polar coordinate plane, a ray is drawn from the origin marking 7π/6 and a point is drawn when this line crosses the circle with radius 0, that is, it marks the origin.$

4) $\left(−4,\frac{3 \pi }{4}\right)$

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

Answer: $On the polar coordinate plane, a ray is drawn from the origin marking π/4 and a point is drawn when this line crosses the circle with radius 1.$

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

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

Answer: $On the polar coordinate plane, a ray is drawn from the origin marking π/2 and a point is drawn when this line crosses the circle with radius 1.$

In exercises 8 - 11, 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.

Answer: $B\left(3,\frac{− \pi }{3}\right) B\left(−3,\frac{2 \pi }{3}\right)$

10) Coordinates of point C.

11) Coordinates of point D.

Answer: $D\left(5,\frac{7 \pi }{6}\right) D\left(−5,\frac{ \pi }{6}\right)$

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

12) $(2,2)$

13) $(3,−4)$

Answer: $(5,−0.927),\;(−5,−0.927+ \pi )$

14) $(8,15)$

15) $(−6,8)$

Answer: $(10,−0.927),\;(−10,−0.927+ \pi )$

16) $(4,3)$

17) $(3,−\sqrt{3})$

Answer: $(2\sqrt{3},−0.524),\;(−2\sqrt{3},−0.524+ \pi )$

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

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

19) $\left(−2,\frac{ \pi }{6}\right)$

Answer: $(−\sqrt{3},−1)$

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

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

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

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

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

Answer: $(0,0)$

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

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

26) $r^2=9\cos \theta $

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

Answer: Symmetric with respect to x-axis only.

28) $r=2\sec \theta $

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

Answer: 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) $ \theta =\frac{ \pi }{4}$

Answer: Line $y=x$

32) $r=\sec \theta $

33) $r=\csc \theta $

Answer: $y=1$

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

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

35) $x^2−y^2=16$

Answer

Hyperbola; polar form $r^2\cos(2 \theta )=16$ 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) $x=8$

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

37) $3x−y=2$

Answer

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

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

38) $y^2=4x$

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

39) $r=4\sin \theta $

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

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

41) $r=6\cos \theta $

42) $r= \theta $

Answer

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

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

43) $r=\cot \theta \csc \theta $

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

44) $r=1+\sin \theta $

Answer

$y$-axis symmetry

$A cardioid with the upper heart part at the origin and the rest of the cardioid oriented up.$

45) $r=3−2\cos \theta $

46) $r=2−2\sin \theta $

Answer

$y$-axis symmetry

$A cardioid with the upper heart part at the origin and the rest of the cardioid oriented down.$

47) $r=5−4\sin \theta $

48) $r=3\cos(2 \theta )$

Answer

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

$A rose with four petals that reach their furthest extent from the origin at θ = 0, π/2, π, and 3π/2.$

49) $r=3\sin(2 \theta )$

50) $r=2\cos(3 \theta )$

Answer: $x$-axis symmetry
$A rose with three petals that reach their furthest extent from the origin at θ = 0, 2π/3, and 4π/3.$

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

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

Answer

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

$The infinity symbol with the crossing point at the origin and with the furthest extent of the two petals being at θ = 0 and π.$

53) $r^2=4\sin \theta $

54) $r=2 \theta $

Answer: no symmetry
$A spiral that starts at the origin crossing the line θ = π/2 between 3 and 4, θ = π between 6 and 7, θ = 3π/2 between 9 and 10, θ = 0 between 12 and 13, θ = π/2 between 15 and 16, and θ = π between 18 and 19.$

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

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

Answer: a line
$A line that crosses the y axis at roughly 3 and has slope roughly 3/2.$

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

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

Answer: $A geometric shape that resembles a butterfly with larger wings in the first and second quadrants, smaller wings in the third and fourth quadrants, a body along the θ = π/2 line and legs along the θ = 0 and π lines.$

59) [Technology Required] Use technology to plot $r=\sin(\frac{3 \theta }{7})$ (use the interval $0 \leq \theta \leq 14 \pi $).

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

Answer: $A line with θ = 120°.$

61) [Technology Required] Use a graphing utility to plot $r= \theta \sin \theta $ for $− \pi \leq \theta \leq \pi $.

62) [Technology Required] Use technology to plot $r=e^{−0.1 \theta }$ for $−10 \leq \theta \leq 10.$

Answer: $A spiral that starts in the third quadrant.$

63) [Technology Required] There is a curve known as the “Black Hole.” Use technology to plot $r=e^{−0.01 \theta }$ for $−100 \leq \theta \leq 100$.

64) [Technology Required] 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 $| \theta |>100$.

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

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