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 is the figure has rectangular coordinates (_____, _____) and polar coordinates (_____, _____).
(GRAPH CAN'T COPY)

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 $x=$ _________ and $y=$ _________.
(b) If $P$ has rectangular coordinates $(x, y)$ then it has polar coordinates $(r, \theta)$ where $r^{2}=$ _________ and $\tan \theta=$ _________.

Anthony Pisaturo

Anthony Pisaturo

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

Yes or No? If No, give a reason.
Do the equations relating polar and rectangular coordinates uniquely determine $r$ and $\theta ?$

Anthony Pisaturo

Anthony Pisaturo

Numerade Educator

Problem 5

Plot the point that has the given polar coordinates.
$$(4, \pi / 4)$$

Joseph Lentino

Numerade Educator

Problem 6

Plot the point that has the given polar coordinates.
$$(1,0)$$

Joseph Lentino

Numerade Educator

Problem 7

Plot the point that has the given polar coordinates.
$$(6,-7 \pi / 6)$$

Joseph Lentino

Numerade Educator

Problem 8

Plot the point that has the given polar coordinates.
$$(3,-2 \pi / 3)$$

Joseph Lentino

Numerade Educator

Problem 9

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

Joseph Lentino

Numerade Educator

Problem 10

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

Joseph Lentino

Numerade Educator

Problem 11

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 $r>0$
$$(3, \pi / 2)$$

Joseph Lentino

Numerade Educator

Problem 12

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 $r>0$
$$(2,3 \pi / 4)$$

Joseph Lentino

Numerade Educator

Problem 13

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 $r>0$
$$(-1,7 \pi / 6)$$

Joseph Lentino

Numerade Educator

Problem 14

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 $r>0$
$$(-2,-\pi / 3)$$

Joseph Lentino

Numerade Educator

Problem 15

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 $r>0$
$$(-5,0)$$

Joseph Lentino

Numerade Educator

Problem 16

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 $r>0$
$$(3,1)$$

Joseph Lentino

Numerade Educator

Problem 17

Determine which point in the figure, $P, Q, R,$ or $S,$ has the given polar coordinates.
(GRAPH CAN'T COPY)
$$(4,3 \pi / 4)$$

Stephen Hobbs

Numerade Educator

Problem 18

Determine which point in the figure, $P, Q, R,$ or $S,$ has the given polar coordinates.
(GRAPH CAN'T COPY)
$$(4,-3 \pi / 4)$$

Stephen Hobbs

Numerade Educator

Problem 19

Determine which point in the figure, $P, Q, R,$ or $S,$ has the given polar coordinates.
(GRAPH CAN'T COPY)
$$(-4,-\pi / 4)$$

Stephen Hobbs

Numerade Educator

Problem 20

Determine which point in the figure, $P, Q, R,$ or $S,$ has the given polar coordinates.
(GRAPH CAN'T COPY)
$$(-4,13 \pi / 4)$$

Stephen Hobbs

Numerade Educator

Problem 21

Determine which point in the figure, $P, Q, R,$ or $S,$ has the given polar coordinates.
(GRAPH CAN'T COPY)
$$(4,-23 \pi / 4)$$

Madi Sousa

Numerade Educator

Problem 22

Determine which point in the figure, $P, Q, R,$ or $S,$ has the given polar coordinates.
(GRAPH CAN'T COPY)
$$(-4,23 \pi / 4)$$

Mary Brese

Numerade Educator

Problem 23

Determine which point in the figure, $P, Q, R,$ or $S,$ has the given polar coordinates.
(GRAPH CAN'T COPY)
$$(-4,101 \pi / 4)$$

Stephen Hobbs

Numerade Educator

Problem 24

Determine which point in the figure, $P, Q, R,$ or $S,$ has the given polar coordinates.
(GRAPH CAN'T COPY)
$$(4,103 \pi / 4)$$

Mary Brese

Numerade Educator

Problem 25

A point is graphed in rectangular form. Find polar coordinates for the point, with $r>0$ and $0<\theta<2 \pi$
(GRAPH CAN'T COPY)

Kevin Harmer

Numerade Educator

Problem 26

A point is graphed in rectangular form. Find polar coordinates for the point, with $r>0$ and $0<\theta<2 \pi$
(GRAPH CAN'T COPY)

Kevin Harmer

Numerade Educator

Problem 27

Polar Coordinates to Rectangular Coordinates A point is graphed in polar form. Find its rectangular coordinates.
(GRAPH CAN'T COPY)

Kevin Harmer

Numerade Educator

Problem 28

Polar Coordinates to Rectangular Coordinates A point is graphed in polar form. Find its rectangular coordinates.
(GRAPH CAN'T COPY)

Kevin Harmer

Numerade Educator

Problem 29

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

Aditya Sood

Numerade Educator

Problem 30

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

Anthony Pisaturo

Anthony Pisaturo

Numerade Educator

Problem 31

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

Aditya Sood

Numerade Educator

Problem 32

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

MP

Michael Pantoja

Numerade Educator

Problem 33

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

Aditya Sood

Numerade Educator

Problem 34

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

Lee Barrow

Numerade Educator

Problem 35

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

Aditya Sood

Numerade Educator

Problem 36

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

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

Kevin Harmer

Numerade Educator

Problem 38

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

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

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

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

Kevin Harmer

Numerade Educator

Problem 42

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

Kevin Harmer

Numerade Educator

Problem 43

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

Kevin Harmer

Numerade Educator

Problem 44

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

Anthony Pisaturo

Anthony Pisaturo

Numerade Educator

Problem 45

Convert the equation to polar form.
$$x=y$$

Joseph Lentino

Numerade Educator

Problem 46

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

Joseph Lentino

Numerade Educator

Problem 47

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

Joseph Lentino

Numerade Educator

Problem 48

Convert the equation to polar form.
$$y=5$$

Joseph Lentino

Numerade Educator

Problem 49

Convert the equation to polar form.
$$x=4$$

Joseph Lentino

Numerade Educator

Problem 50

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

Joseph Lentino

Numerade Educator

Problem 51

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

Aditya Sood

Numerade Educator

Problem 52

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

Anthony Pisaturo

Anthony Pisaturo

Numerade Educator

Problem 53

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

Aditya Sood

Numerade Educator

Problem 54

Convert the polar equation to rectangular coordinates.
$$\boldsymbol{\theta}=\pi$$

Joseph Lentino

Numerade Educator

Problem 55

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

Khushbu Rani

Numerade Educator

Problem 56

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

Anthony Pisaturo

Anthony Pisaturo

Numerade Educator

Problem 57

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

Kevin Harmer

Numerade Educator

Problem 58

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

Khushbu Rani

Numerade Educator

Problem 59

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

Kevin Harmer

Numerade Educator

Problem 60

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

Anthony Pisaturo

Anthony Pisaturo

Numerade Educator

Problem 61

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

Kevin Harmer

Numerade Educator

Problem 62

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

Anthony Pisaturo

Anthony Pisaturo

Numerade Educator

Problem 63

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

Kevin Harmer

Numerade Educator

Problem 64

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

Anthony Pisaturo

Anthony Pisaturo

Numerade Educator

Problem 65

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

Kevin Harmer

Numerade Educator

Problem 66

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

Anthony Pisaturo

Anthony Pisaturo

Numerade Educator

Problem 67

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

Kevin Harmer

Numerade Educator

Problem 68

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.
$$\sec \theta=2$$

Joseph Lentino

Numerade Educator

Problem 70

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

Anthony Pisaturo

Anthony Pisaturo

Numerade Educator

Problem 71

(a) Use the Law of cosines to prove that the distance between the polar points $\left(r_{1}, \theta_{1}\right)$ and $\left(r_{2}, \theta_{2}\right)$ is
$$
d=\sqrt{r_{1}^{2}+r_{2}^{2}-2 r_{1} r_{2} \cos \left(\theta_{2}-\theta_{1}\right)}
$$
(b) Find the distance between the points whose polar coordinates are $(3,3 \pi / 4)$ and $(1,7 \pi / 6),$ using the formula from part (a).
(c) Now convert the points in part (b) to rectangular coordinates. Find the distance between them, using the usual Distance Formula. Do you get the same answer?

Sandro Maludze

Numerade Educator

Problem 72

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.
(FIGURE CAN'T COPY)

Anthony Pisaturo

Anthony Pisaturo

Numerade Educator