Precalculus Mathematics for Calculus

James Stewart, Lothar Redlin, Saleem Watson

Chapter 8

Polar Coordinates and Parametric Equations - all with Video Answers

Educators

Section 1

Polar Coordinates

Problem 1

We can describe the location of a point in the plane using different_________________systems. The point $P$ shown in the figure has rectangular coordinates $(\quad, \quad)$ and polar
coordinates $(\quad, \quad)$

Aditya Sood

Aditya Sood

Numerade Educator

Problem 2

Let $P$ be a point in the plane.
(a) If $P$ has polar coordinates $(r, \theta)$ then it has rectangular coordinates $(x, y)$ where = _____________and $y=$________________
(b) If $P$ has rectangular coordinates $(x, y)$ then it has polar coordinates $(r, \theta)$ where $r^{2}=$ _____________and $\tan \theta=$_____________

Nathan Mankovich

Nathan Mankovich

Numerade Educator

Problem 3

Yes or No? If No, give a reason.
Do the polar coordinates $(2, \pi / 6)$ and $(-2,7 \pi / 6)$ represent
the same point?

Nathan Mankovich

Nathan Mankovich

Numerade Educator

Problem 4

Do the equations relating polar and rectangular coordinates
uniquely determine $r$ and $\theta$ ?

Nathan Mankovich

Nathan Mankovich

Numerade Educator

Problem 5

Plotting Points in Polar Coordinates Plot the point that
has the given polar coordinates.
$(4, \pi / 4)$

Nathan Mankovich

Nathan Mankovich

Numerade Educator

Problem 6

Plotting Points in Polar Coordinates Plot the point that has the given polar coordinates. $(1,0)$

Nathan Mankovich

Nathan Mankovich

Numerade Educator

Problem 7

Plotting Points in Polar Coordinates Plot the point that has the given polar coordinates. $(6,-7 \pi / 6)$

Nathan Mankovich

Nathan Mankovich

Numerade Educator

Problem 8

Plotting Points in Polar Coordinates Plot the point that has the given polar coordinates. $(3,-2 \pi / 3)$

Nathan Mankovich

Nathan Mankovich

Numerade Educator

Problem 9

Plotting Points in Polar Coordinates Plot the point that has the given polar coordinates. $(-2,4 \pi / 3)$

Nathan Mankovich

Nathan Mankovich

Numerade Educator

Problem 10

Plotting Points in Polar Coordinates Plot the point that has the given polar coordinates. $(-5,-17 \pi / 6)$

Nathan Mankovich

Nathan Mankovich

Numerade Educator

Problem 11

Different Polar Coordinates for the Same Point Plot the point that has the given polar coordinates. Then give two other
polar coordinate representations of the point, one with $r<0$ and
the other with $r>0 .$ $(3, \pi / 2)$

Nathan Mankovich

Nathan Mankovich

Numerade Educator

Problem 12

Different Polar Coordinates for the Same Point Plot the point that has the given polar coordinates. Then give two other
polar coordinate representations of the point, one with $r<0$ and
the other with $r>0 .$ $(2,3 \pi / 4)$

Nathan Mankovich

Nathan Mankovich

Numerade Educator

Problem 13

Different Polar Coordinates for the Same Point Plot the point that has the given polar coordinates. Then give two other
polar coordinate representations of the point, one with $r<0$ and
the other with $r>0 .$ $(-1,7 \pi / 6)$

Nathan Mankovich

Nathan Mankovich

Numerade Educator

Problem 14

Different Polar Coordinates for the Same Point Plot the point that has the given polar coordinates. Then give two other
polar coordinate representations of the point, one with $r<0$ and
the other with $r>0 .$ $(-2,-\pi / 3)$

Nathan Mankovich

Nathan Mankovich

Numerade Educator

Problem 15

Different Polar Coordinates for the Same Point Plot the point that has the given polar coordinates. Then give two other
polar coordinate representations of the point, one with $r<0$ and
the other with $r>0 .$ $(-5,0)$

Kevin Harmer

Numerade Educator

Problem 16

Different Polar Coordinates for the Same Point Plot the point that has the given polar coordinates. Then give two other
polar coordinate representations of the point, one with $r<0$ and
the other with $r>0 .$ $(3,1)$

Anthony Pisaturo

Anthony Pisaturo

Numerade Educator

Problem 17

Points in Polar Coordinates Determine which point in the figure, $P, Q, R,$ or $S,$ has the given polar coordinates.
$(4,3 \pi / 4)$

Aditya Sood

Numerade Educator

Problem 18

Points in Polar Coordinates Determine which point in the figure, $P, Q, R,$ or $S,$ has the given polar coordinates.
$(4,-3 \pi / 4)$

Suman Saurav Thakur

Suman Saurav Thakur

Numerade Educator

Problem 19

Points in Polar Coordinates Determine which point in the figure, $P, Q, R,$ or $S,$ has the given polar coordinates.
$(-4,-\pi / 4)$

Aditya Sood

Numerade Educator

Problem 20

Points in Polar Coordinates Determine which point in the figure, $P, Q, R,$ or $S,$ has the given polar coordinates.
$(-4,13 \pi / 4)$

Anthony Pisaturo

Anthony Pisaturo

Numerade Educator

Problem 21

Points in Polar Coordinates Determine which point in the figure, $P, Q, R,$ or $S,$ has the given polar coordinates.
$(4,-23 \pi / 4)$

Aditya Sood

Numerade Educator

Problem 22

Points in Polar Coordinates Determine which point in the figure, $P, Q, R,$ or $S,$ has the given polar coordinates.
$(-4,23 \pi / 4)$

Anthony Pisaturo

Anthony Pisaturo

Numerade Educator

Problem 23

Points in Polar Coordinates Determine which point in the figure, $P, Q, R,$ or $S,$ has the given polar coordinates.
$(-4,101 \pi / 4)$

Aditya Sood

Numerade Educator

Problem 24

Points in Polar Coordinates Determine which point in the figure, $P, Q, R,$ or $S,$ has the given polar coordinates.
$(4,103 \pi / 4)$

Anthony Pisaturo

Anthony Pisaturo

Numerade Educator

Problem 25

Rectangular Coordinates to Polar Coordinates A point is graphed in rectangular form. Find polar coordinates for the
point, with $r>0$ and $0<\theta<2 \pi$.
(Graph cant copy)

Kevin Harmer

Numerade Educator

Problem 26

Rectangular Coordinates to Polar Coordinates A point is graphed in rectangular form. Find polar coordinates for the
point, with $r>0$ and $0<\theta<2 \pi$.
(Graph cant copy)

Kevin Harmer

Numerade Educator

Problem 27

Polar Coordinates to Rectangular Coordinates A point is graphed in polar form. Find its rectangular coordinates.
(Graph cant copy)

Kevin Harmer

Numerade Educator

Problem 28

Polar Coordinates to Rectangular Coordinates A point is graphed in polar form. Find its rectangular coordinates.
(Graph cant copy)

Kevin Harmer

Numerade Educator

Problem 29

Polar Coordinates to Rectangular Coordinates Find the rectangular coordinates for the point whose polar coordinates are given.
$(4, \pi / 6)$

Aditya Sood

Numerade Educator

Problem 30

Polar Coordinates to Rectangular Coordinates Find the rectangular coordinates for the point whose polar coordinates are given. $(6,2 \pi / 3)$

Anthony Pisaturo

Anthony Pisaturo

Numerade Educator

Problem 31

Polar Coordinates to Rectangular Coordinates Find the rectangular coordinates for the point whose polar coordinates are given. $(\sqrt{2},-\pi / 4)$

Aditya Sood

Numerade Educator

Problem 32

Polar Coordinates to Rectangular Coordinates Find the rectangular coordinates for the point whose polar coordinates are given. $(-1,5 \pi / 2)$

Anthony Pisaturo

Anthony Pisaturo

Numerade Educator

Problem 33

Polar Coordinates to Rectangular Coordinates Find the rectangular coordinates for the point whose polar coordinates are given. $(5,5 \pi)$

Aditya Sood

Numerade Educator

Problem 34

Polar Coordinates to Rectangular Coordinates Find the rectangular coordinates for the point whose polar coordinates are given. $(0,13 \pi)$

Anthony Pisaturo

Anthony Pisaturo

Numerade Educator

Problem 35

Polar Coordinates to Rectangular Coordinates Find the rectangular coordinates for the point whose polar coordinates are given. $(6 \sqrt{2}, 11 \pi / 6)$

Aditya Sood

Numerade Educator

Problem 36

Polar Coordinates to Rectangular Coordinates Find the rectangular coordinates for the point whose polar coordinates are given. $(\sqrt{3},-5 \pi / 3)$

Anthony Pisaturo

Anthony Pisaturo

Numerade Educator

Problem 37

Rectangular Coordinates to Polar Coordinates Convert the rectangular coordinates to polar coordinates with $r>0$ and
$0 \leq \theta<2 \pi$.
$(-1,1)$

Kevin Harmer

Numerade Educator

Problem 38

Rectangular Coordinates to Polar Coordinates Convert the rectangular coordinates to polar coordinates with $r>0$ and
$0 \leq \theta<2 \pi$.
$$
(3 \sqrt{3},-3)
$$

Kevin Harmer

Numerade Educator

Problem 39

Rectangular Coordinates to Polar Coordinates Convert the rectangular coordinates to polar coordinates with $r>0$ and
$0 \leq \theta<2 \pi$.
$$
(\sqrt{8}, \sqrt{8})
$$

Kevin Harmer

Numerade Educator

Problem 40

Rectangular Coordinates to Polar Coordinates Convert the rectangular coordinates to polar coordinates with $r>0$ and
$0 \leq \theta<2 \pi$.
$$
(-\sqrt{6},-\sqrt{2})
$$

Kevin Harmer

Numerade Educator

Problem 41

Rectangular Coordinates to Polar Coordinates Convert the rectangular coordinates to polar coordinates with $r>0$ and
$0 \leq \theta<2 \pi$.
$$
(3,4)
$$

Kevin Harmer

Numerade Educator

Problem 42

Rectangular Coordinates to Polar Coordinates Convert the rectangular coordinates to polar coordinates with $r>0$ and
$0 \leq \theta<2 \pi$.
$$
(1,-2)
$$

Kevin Harmer

Numerade Educator

Problem 43

Rectangular Coordinates to Polar Coordinates Convert the rectangular coordinates to polar coordinates with $r>0$ and
$0 \leq \theta<2 \pi$.
$$
(-6,0)
$$

Kevin Harmer

Numerade Educator

Problem 44

Rectangular Equations to Polar Equations Convert the equation to polar form.
$$
(0,-\sqrt{3})
$$

Kevin Harmer

Numerade Educator

Problem 45

Rectangular Equations to Polar Equations Convert the equation to polar form.
$$
x=y
$$

Aditya Sood

Numerade Educator

Problem 46

Rectangular Equations to Polar Equations Convert the equation to polar form.
$$
x^{2}+y^{2}=9
$$

Anthony Pisaturo

Anthony Pisaturo

Numerade Educator

Problem 47

Rectangular Equations to Polar Equations Convert the equation to polar form.
$$
y=x^{2}
$$

Kevin Harmer

Numerade Educator

Problem 48

Rectangular Equations to Polar Equations Convert the equation to polar form.
$$
y=5
$$

Anthony Pisaturo

Anthony Pisaturo

Numerade Educator

Problem 49

Rectangular Equations to Polar Equations Convert the equation to polar form.
$$
x=4
$$

Aditya Sood

Numerade Educator

Problem 50

Rectangular Equations to Polar Equations Convert the equation to polar form.
$$
x^{2}-y^{2}=1
$$

Anthony Pisaturo

Anthony Pisaturo

Numerade Educator

Problem 51

Polar Equations to Rectangular Equations Convert the polar equation to rectangular coordinates.
$$
r=7
$$

Aditya Sood

Numerade Educator

Problem 52

Polar Equations to Rectangular Equations Convert the polar equation to rectangular coordinates.
$$
r=-3
$$

Anthony Pisaturo

Anthony Pisaturo

Numerade Educator

Problem 53

Polar Equations to Rectangular Equations Convert the polar equation to rectangular coordinates.
$$
\theta=-\frac{\pi}{2}
$$

Aditya Sood

Numerade Educator

Problem 54

Polar Equations to Rectangular Equations Convert the polar equation to rectangular coordinates.
$$
\theta=\pi
$$

Anthony Pisaturo

Anthony Pisaturo

Numerade Educator

Problem 55

Polar Equations to Rectangular Equations Convert the polar equation to rectangular coordinates.
$$
r \cos \theta=6
$$

Anthony Pisaturo

Anthony Pisaturo

Numerade Educator

Problem 56

Polar Equations to Rectangular Equations Convert the polar equation to rectangular coordinates.
$$
r=2 \csc \theta
$$

Anthony Pisaturo

Anthony Pisaturo

Numerade Educator

Problem 57

Polar Equations to Rectangular Equations Convert the polar equation to rectangular coordinates.
$$
r=4 \sin \theta
$$

Kevin Harmer

Numerade Educator

Problem 58

Polar Equations to Rectangular Equations Convert the polar equation to rectangular coordinates.
$$
r=6 \cos \theta
$$

Aditya Sood

Numerade Educator

Problem 59

Polar Equations to Rectangular Equations Convert the polar equation to rectangular coordinates.
$$
r=1+\cos \theta
$$

Kevin Harmer

Numerade Educator

Problem 60

Polar Equations to Rectangular Equations Convert the polar equation to rectangular coordinates.
$$
r=3(1-\sin \theta)
$$

Anthony Pisaturo

Anthony Pisaturo

Numerade Educator

Problem 61

Polar Equations to Rectangular Equations Convert the polar equation to rectangular coordinates.
$$
r=1+2 \sin \theta
$$

Kevin Harmer

Numerade Educator

Problem 62

Polar Equations to Rectangular Equations Convert the polar equation to rectangular coordinates.
$$
r=2-\cos \theta
$$

Anthony Pisaturo

Anthony Pisaturo

Numerade Educator

Problem 63

Polar Equations to Rectangular Equations Convert the polar equation to rectangular coordinates.
$$
r=\frac{1}{\sin \theta-\cos \theta}
$$

Kevin Harmer

Numerade Educator

Problem 64

Polar Equations to Rectangular Equations Convert the polar equation to rectangular coordinates.
$$
r=\frac{1}{1+\sin \theta}
$$

Anthony Pisaturo

Anthony Pisaturo

Numerade Educator

Problem 65

Polar Equations to Rectangular Equations Convert the polar equation to rectangular coordinates.
$$
r=\frac{4}{1+2 \sin \theta}
$$

Kevin Harmer

Numerade Educator

Problem 66

Polar Equations to Rectangular Equations Convert the polar equation to rectangular coordinates.
$$
r=\frac{2}{1-\cos \theta}
$$

Anthony Pisaturo

Anthony Pisaturo

Numerade Educator

Problem 67

Polar Equations to Rectangular Equations Convert the polar equation to rectangular coordinates.
$$
r^{2}=\tan \theta
$$

Kevin Harmer

Numerade Educator

Problem 68

Polar Equations to Rectangular Equations Convert the polar equation to rectangular coordinates.
$$
r^{2}=\sin 2 \theta
$$

Anthony Pisaturo

Anthony Pisaturo

Numerade Educator

Problem 69

Convert the polar equation to rectangular coordinates.
$$
\sec \theta=2
$$

Kevin Harmer

Numerade Educator

Problem 70

Polar Equations to Rectangular Equations Convert the polar equation to rectangular coordinates.
$$
\cos 2 \theta=1
$$

Anthony Pisaturo

Anthony Pisaturo

Numerade Educator

Problem 71

Discuss a pRoviE: The Distance Formula in Polar Coordinates
$$
\begin{array}{l}{\text { (a) Use the Law of Cosines to prove that the distance }} \\ {\text { between the polar points }\left(r_{1}, \theta_{1}\right) \text { and }\left(r_{2}, \theta_{2}\right) \text { is }} \\ {\qquad d=\sqrt{r_{1}^{2}+r_{2}^{2}-2 r_{1} r_{2} \cos \left(\theta_{2}-\theta_{1}\right)}} \\ {\text { (b) Find the distance between the points whose polar coordi- }} \\ {\text { nates are }(3,3 \pi / 4) \text { and }(1,7 \pi / 6), \text { using the formula }} \\ {\text { from part ( a). }} \\ {\text { (c) Now convert the points in part (b) to rectangular coordi- }} \\ {\text { nates. Find the distance between them, using the usual }} \\ {\text { Distance Formula. Do you get the same answer? }}\end{array}
$$

Sandro Maludze

Numerade Educator

Problem 72

DISCUSS: Different Coordinate Systems As was noted in the overview of the chapter, certain curves are more naturally described in one coordinate system than in another. In each of the following situations, which coordinate system would be appropriate: rectangular or polar? Give reasons to support your answer.
(a) You need to give directions to your house to a taxi driver.
(b) You need to give directions to your house to a homing
pigeon.

Anthony Pisaturo

Anthony Pisaturo

Numerade Educator