• Home
  • Textbooks
  • Precalculus
  • Polar Coordinates; Vectors

Precalculus

Michael Sullivan

Chapter 9

Polar Coordinates; Vectors - all with Video Answers

Educators

Section 1

Polar Coordinates

00:37

Problem 1

Plot the point whose rectangular coordinates are (3,-1) . What quadrant does the point lie in?

Katelyn Chen
Katelyn Chen
Numerade Educator
00:23

Problem 2

To complete the square of $x^{2}+6 x,$ add ____.

Katelyn Chen
Katelyn Chen
Numerade Educator
00:25

Problem 3

If $P=(a, b)$ is a point on the terminal side of the angle $\theta$ at a distance $r$ from the origin, then $\tan \theta=$ ______.

Katelyn Chen
Katelyn Chen
Numerade Educator
00:12

Problem 4

$\tan ^{-1}(-1)=$ ____.

Katelyn Chen
Katelyn Chen
Numerade Educator
01:08

Problem 5

The origin in rectangular coordinates coincides with the ____ in polar coordinates; the positive $x$ -axis in rectangular coordinates coincides with the ____ ____ in polar coordinates.

Katelyn Chen
Katelyn Chen
Numerade Educator
00:39

Problem 6

If $P$ is a point with polar coordinates $(r, \theta),$ the rectangular coordinates $(x, y)$ of $P$ are given by $x=$ ____ and $y=$ ____.

Katelyn Chen
Katelyn Chen
Numerade Educator
00:36

Problem 7

For the point with polar coordinates $\left(1,-\frac{\pi}{2}\right),$ which of
the following best describes the location of the point in a rectangular coordinate system?
(a) in quadrant IV
(b) on the $y$ -axis
(c) in quadrant II
(d) on the $x$ -axis

Katelyn Chen
Katelyn Chen
Numerade Educator
01:10

Problem 8

The point $\left(5, \frac{\pi}{6}\right)$ can also be represented by which of the following polar coordinates?
(a) $\left(5,-\frac{\pi}{6}\right)$
(b) $\left(-5, \frac{13 \pi}{6}\right)$
(c) $\left(5,-\frac{5 \pi}{6}\right)$
(d) $\left(-5, \frac{7 \pi}{6}\right)$

Katelyn Chen
Katelyn Chen
Numerade Educator
00:31

Problem 9

True or False
In the polar coordinates $(r, \theta), r$ can be negative.

Katelyn Chen
Katelyn Chen
Numerade Educator
00:16

Problem 10

True or False
The polar coordinates of a point are unique.

Katelyn Chen
Katelyn Chen
Numerade Educator
00:40

Problem 11

Match each point in polar coordinates with either A, B, C, or D on the graph.
$$\left(2,-\frac{11 \pi}{6}\right)$$

Aamir Mithaiwala
Aamir Mithaiwala
Numerade Educator
00:37

Problem 12

Match each point in polar coordinates with either A, B, C, or D on the graph.
$$\left(-2,-\frac{\pi}{6}\right)$$

Aamir Mithaiwala
Aamir Mithaiwala
Numerade Educator
00:40

Problem 13

Match each point in polar coordinates with either A, B, C, or D on the graph.
$$\left(-2, \frac{\pi}{6}\right)$$

Aamir Mithaiwala
Aamir Mithaiwala
Numerade Educator
00:40

Problem 14

Match each point in polar coordinates with either A, B, C, or D on the graph.
$$\left(2, \frac{7 \pi}{6}\right)$$

Aamir Mithaiwala
Aamir Mithaiwala
Numerade Educator
00:42

Problem 15

Match each point in polar coordinates with either A, B, C, or D on the graph.
$$\left(2, \frac{5 \pi}{6}\right)$$

Aamir Mithaiwala
Aamir Mithaiwala
Numerade Educator
00:36

Problem 16

Match each point in polar coordinates with either A, B, C, or D on the graph.
$$\left(-2, \frac{5 \pi}{6}\right)$$

Aamir Mithaiwala
Aamir Mithaiwala
Numerade Educator
00:36

Problem 17

Match each point in polar coordinates with either A, B, C, or D on the graph.
$$\left(-2, \frac{7 \pi}{6}\right)$$

Aamir Mithaiwala
Aamir Mithaiwala
Numerade Educator
01:05

Problem 18

Match each point in polar coordinates with either A, B, C, or D on the graph.
$$\left(2, \frac{11 \pi}{6}\right)$$

Aamir Mithaiwala
Aamir Mithaiwala
Numerade Educator
01:52

Problem 19

Plot each point given in polar coordinates.
$$\left(3,90^{\circ}\right)$$

Aamir Mithaiwala
Aamir Mithaiwala
Numerade Educator
00:59

Problem 20

Plot each point given in polar coordinates.
$$\left(4,270^{\circ}\right)$$

Aamir Mithaiwala
Aamir Mithaiwala
Numerade Educator
00:43

Problem 21

Plot each point given in polar coordinates.
$$(-2,0)$$

Aamir Mithaiwala
Aamir Mithaiwala
Numerade Educator
01:05

Problem 22

Plot each point given in polar coordinates.
$$
(-3, \pi)
$$

Aamir Mithaiwala
Aamir Mithaiwala
Numerade Educator
01:05

Problem 23

Plot each point given in polar coordinates.
$$(-3, \pi)$$

Aamir Mithaiwala
Aamir Mithaiwala
Numerade Educator
01:49

Problem 24

Plot each point given in polar coordinates.
$$\left(5, \frac{5 \pi}{3}\right)$$

Aamir Mithaiwala
Aamir Mithaiwala
Numerade Educator
01:53

Problem 25

Plot each point given in polar coordinates.
$$\left(-2,135^{\circ}\right)$$

Aamir Mithaiwala
Aamir Mithaiwala
Numerade Educator
01:35

Problem 26

Plot each point given in polar coordinates.
$$\left(-3,120^{\circ}\right)$$

Aamir Mithaiwala
Aamir Mithaiwala
Numerade Educator
01:28

Problem 27

Plot each point given in polar coordinates.
$$\left(4,-\frac{2 \pi}{3}\right)$$

Aamir Mithaiwala
Aamir Mithaiwala
Numerade Educator
01:59

Problem 28

Plot each point given in polar coordinates.
$$\left(2,-\frac{5 \pi}{4}\right)$$

Aamir Mithaiwala
Aamir Mithaiwala
Numerade Educator
02:02

Problem 29

Plot each point given in polar coordinates.
$$\left(-1,-\frac{\pi}{3}\right)$$

Aamir Mithaiwala
Aamir Mithaiwala
Numerade Educator
02:03

Problem 30

Plot each point given in polar coordinates.
$$\left(-3,-\frac{3 \pi}{4}\right)$$

Aamir Mithaiwala
Aamir Mithaiwala
Numerade Educator
01:33

Problem 31

Plot each point given in polar coordinates.
$$(-2,-\pi)$$

Aamir Mithaiwala
Aamir Mithaiwala
Numerade Educator
01:39

Problem 32

Plot each point given in polar coordinates.
$$\left(-3,-\frac{\pi}{2}\right)$$

Aamir Mithaiwala
Aamir Mithaiwala
Numerade Educator
05:20

Problem 33

Plot each point given in polar coordinates, and find other polar coordinates $(r, \theta)$ of the point for which:
(a) $r>0, \quad-2 \pi \leq \theta<0$
(b) $r<0, \quad 0 \leq \theta<2 \pi$
(c) $r>0, \quad 2 \pi \leq \theta<4 \pi$e$\left(5, \frac{2 \pi}{3}\right)$

Chris Wojturski
Chris Wojturski
Numerade Educator
06:02

Problem 34

Plot each point given in polar coordinates, and find other polar coordinates $(r, \theta)$ of the point for which:
(a) $r>0, \quad-2 \pi \leq \theta<0$
(b) $r<0, \quad 0 \leq \theta<2 \pi$
(c) $r>0, \quad 2 \pi \leq \theta<4 \pi$e
$\left(4, \frac{3 \pi}{4}\right)$

Chris Wojturski
Chris Wojturski
Numerade Educator
05:47

Problem 35

Plot each point given in polar coordinates, and find other polar coordinates $(r, \theta)$ of the point for which:
(a) $r>0, \quad-2 \pi \leq \theta<0$
(b) $r<0, \quad 0 \leq \theta<2 \pi$
(c) $r>0, \quad 2 \pi \leq \theta<4 \pi$e
$$(-2,3 \pi)$$

Chris Wojturski
Chris Wojturski
Numerade Educator
05:42

Problem 36

Plot each point given in polar coordinates, and find other polar coordinates $(r, \theta)$ of the point for which:
(a) $r>0, \quad-2 \pi \leq \theta<0$
(b) $r<0, \quad 0 \leq \theta<2 \pi$
(c) $r>0, \quad 2 \pi \leq \theta<4 \pi$e
$$(-3,4 \pi)$$

Chris Wojturski
Chris Wojturski
Numerade Educator
05:07

Problem 37

Plot each point given in polar coordinates, and find other polar coordinates $(r, \theta)$ of the point for which:
(a) $r>0, \quad-2 \pi \leq \theta<0$
(b) $r<0, \quad 0 \leq \theta<2 \pi$
(c) $r>0, \quad 2 \pi \leq \theta<4 \pi$e
$$\left(1, \frac{\pi}{2}\right)$$

Chris Wojturski
Chris Wojturski
Numerade Educator
05:13

Problem 38

Plot each point given in polar coordinates, and find other polar coordinates $(r, \theta)$ of the point for which:
(a) $r>0, \quad-2 \pi \leq \theta<0$
(b) $r<0, \quad 0 \leq \theta<2 \pi$
(c) $r>0, \quad 2 \pi \leq \theta<4 \pi$e
$$(2, \pi)$$

Chris Wojturski
Chris Wojturski
Numerade Educator
06:21

Problem 39

Plot each point given in polar coordinates, and find other polar coordinates $(r, \theta)$ of the point for which:
(a) $r>0, \quad-2 \pi \leq \theta<0$
(b) $r<0, \quad 0 \leq \theta<2 \pi$
(c) $r>0, \quad 2 \pi \leq \theta<4 \pi$e
$$\left(-3,-\frac{\pi}{4}\right)$$

Chris Wojturski
Chris Wojturski
Numerade Educator
05:57

Problem 40

Plot each point given in polar coordinates, and find other polar coordinates $(r, \theta)$ of the point for which:
(a) $r>0, \quad-2 \pi \leq \theta<0$
(b) $r<0, \quad 0 \leq \theta<2 \pi$
(c) $r>0, \quad 2 \pi \leq \theta<4 \pi$e
$$\left(-2,-\frac{2 \pi}{3}\right)$$

Chris Wojturski
Chris Wojturski
Numerade Educator
00:52

Problem 41

Polar coordinates of a point are given. Find the rectangular coordinates of each point.
$$\left(3, \frac{\pi}{2}\right)$$

Aamir Mithaiwala
Aamir Mithaiwala
Numerade Educator
00:47

Problem 42

Polar coordinates of a point are given. Find the rectangular coordinates of each point.
$$\left(4, \frac{3 \pi}{2}\right)$$

Aamir Mithaiwala
Aamir Mithaiwala
Numerade Educator
00:46

Problem 43

Polar coordinates of a point are given. Find the rectangular coordinates of each point.
$$(-2,0)$$

Aamir Mithaiwala
Aamir Mithaiwala
Numerade Educator
00:45

Problem 44

Polar coordinates of a point are given. Find the rectangular coordinates of each point.
$$(-3, \pi)$$

Aamir Mithaiwala
Aamir Mithaiwala
Numerade Educator
01:05

Problem 45

Polar coordinates of a point are given. Find the rectangular coordinates of each point.
$$\left(6,150^{\circ}\right)$$

Aamir Mithaiwala
Aamir Mithaiwala
Numerade Educator
00:52

Problem 46

Polar coordinates of a point are given. Find the rectangular coordinates of each point.
$$\left(5,300^{\circ}\right)$$

Aamir Mithaiwala
Aamir Mithaiwala
Numerade Educator
01:04

Problem 47

Polar coordinates of a point are given. Find the rectangular coordinates of each point.
$$\left(-2, \frac{3 \pi}{4}\right)$$

Aamir Mithaiwala
Aamir Mithaiwala
Numerade Educator
01:04

Problem 48

Polar coordinates of a point are given. Find the rectangular coordinates of each point.
$$\left(-2, \frac{3 \pi}{4}\right)$$

Aamir Mithaiwala
Aamir Mithaiwala
Numerade Educator
01:13

Problem 49

Polar coordinates of a point are given. Find the rectangular coordinates of each point.
$$\left(-1,-\frac{\pi}{3}\right)$$

Aamir Mithaiwala
Aamir Mithaiwala
Numerade Educator
01:14

Problem 50

Polar coordinates of a point are given. Find the rectangular coordinates of each point.
$$\left(-3,-\frac{3 \pi}{4}\right)$$

Aamir Mithaiwala
Aamir Mithaiwala
Numerade Educator
00:50

Problem 51

Polar coordinates of a point are given. Find the rectangular coordinates of each point.
$$\left(-2,-180^{\circ}\right)$$

Aamir Mithaiwala
Aamir Mithaiwala
Numerade Educator
00:56

Problem 52

Polar coordinates of a point are given. Find the rectangular coordinates of each point.
$$\left(-3,-90^{\circ}\right)$$

Aamir Mithaiwala
Aamir Mithaiwala
Numerade Educator
00:56

Problem 52

In Problems 41–56, polar coordinates of a point are given. Find the rectangular coordinates of each point.
$$\left(-3,-90^{\circ}\right)$$

Aamir Mithaiwala
Aamir Mithaiwala
Numerade Educator
01:05

Problem 53

Polar coordinates of a point are given. Find the rectangular coordinates of each point.
$$\left(7.5,110^{\circ}\right)$$

Aamir Mithaiwala
Aamir Mithaiwala
Numerade Educator
00:58

Problem 53

In Problems 41–56, polar coordinates of a point are given. Find the rectangular coordinates of each point.
$$\left(7.5,110^{\circ}\right)$$

Katelyn Chen
Katelyn Chen
Numerade Educator
00:37

Problem 54

In Problems 41–56, polar coordinates of a point are given. Find the rectangular coordinates of each point.
$$\left(-3.1,182^{\circ}\right)$$

Katelyn Chen
Katelyn Chen
Numerade Educator
00:58

Problem 54

Polar coordinates of a point are given. Find the rectangular coordinates of each point.
$$\left(-3.1,182^{\circ}\right)$$

Aamir Mithaiwala
Aamir Mithaiwala
Numerade Educator
01:04

Problem 55

Polar coordinates of a point are given. Find the rectangular coordinates of each point.
$$(6.3,3.8)$$

Aamir Mithaiwala
Aamir Mithaiwala
Numerade Educator
00:56

Problem 56

Polar coordinates of a point are given. Find the rectangular coordinates of each point.
$$(8.1,5.2)$$

Aamir Mithaiwala
Aamir Mithaiwala
Numerade Educator
01:17

Problem 57

The rectangular coordinates of a point are given. Find polar coordinates for each point.
$$(3,0)$$

Aamir Mithaiwala
Aamir Mithaiwala
Numerade Educator
01:07

Problem 58

The rectangular coordinates of a point are given. Find polar coordinates for each point.
$$(0,2)$$

Aamir Mithaiwala
Aamir Mithaiwala
Numerade Educator
01:12

Problem 59

The rectangular coordinates of a point are given. Find polar coordinates for each point.
$$(-1,0)$$

Aamir Mithaiwala
Aamir Mithaiwala
Numerade Educator
01:05

Problem 60

The rectangular coordinates of a point are given. Find polar coordinates for each point.
$$(0,-2)$$

Aamir Mithaiwala
Aamir Mithaiwala
Numerade Educator
01:40

Problem 61

The rectangular coordinates of a point are given. Find polar coordinates for each point.
$(1,-1)$$

Aamir Mithaiwala
Aamir Mithaiwala
Numerade Educator
01:11

Problem 62

The rectangular coordinates of a point are given. Find polar coordinates for each point.
$$(-3,3)$$

Aamir Mithaiwala
Aamir Mithaiwala
Numerade Educator
01:04

Problem 63

The rectangular coordinates of a point are given. Find polar coordinates for each point.
$$(\sqrt{3}, 1)$$

Aamir Mithaiwala
Aamir Mithaiwala
Numerade Educator
01:09

Problem 64

The rectangular coordinates of a point are given. Find polar coordinates for each point.
$$(-2,-2 \sqrt{3})$$

Aamir Mithaiwala
Aamir Mithaiwala
Numerade Educator
01:07

Problem 65

The rectangular coordinates of a point are given. Find polar coordinates for each point.
$$(1.3,-2.1)$$

Aamir Mithaiwala
Aamir Mithaiwala
Numerade Educator
01:13

Problem 66

The rectangular coordinates of a point are given. Find polar coordinates for each point.
$$(-0.8,-2.1)$$

Aamir Mithaiwala
Aamir Mithaiwala
Numerade Educator
01:06

Problem 67

The rectangular coordinates of a point are given. Find polar coordinates for each point.
$$(8.3,4.2)$$

Aamir Mithaiwala
Aamir Mithaiwala
Numerade Educator
01:08

Problem 68

The rectangular coordinates of a point are given. Find polar coordinates for each point.
$$(-2.3,0.2)$$

Aamir Mithaiwala
Aamir Mithaiwala
Numerade Educator
00:58

Problem 69

The letters x and y represent rectangular coordinates. Write each equation using polar coordinates $(r, \theta)$
$$2 x^{2}+2 y^{2}=3$$

Aamir Mithaiwala
Aamir Mithaiwala
Numerade Educator
00:38

Problem 70

The letters x and y represent rectangular coordinates. Write each equation using polar coordinates $(r, \theta)$
$$x^{2}+y^{2}=x$$

Aamir Mithaiwala
Aamir Mithaiwala
Numerade Educator
00:41

Problem 71

The letters x and y represent rectangular coordinates. Write each equation using polar coordinates $(r, \theta)$
$$x^{2}=4 y$$

Aamir Mithaiwala
Aamir Mithaiwala
Numerade Educator
00:45

Problem 72

The letters x and y represent rectangular coordinates. Write each equation using polar coordinates $(r, \theta)$
$$y^{2}=2 x$$

Aamir Mithaiwala
Aamir Mithaiwala
Numerade Educator
00:57

Problem 73

The letters x and y represent rectangular coordinates. Write each equation using polar coordinates $(r, \theta)$
$$2 x y=1$$

Aamir Mithaiwala
Aamir Mithaiwala
Numerade Educator
00:59

Problem 74

The letters x and y represent rectangular coordinates. Write each equation using polar coordinates $(r, \theta)$
$$4 x^{2} y=1$$

Aamir Mithaiwala
Aamir Mithaiwala
Numerade Educator
00:22

Problem 75

The letters x and y represent rectangular coordinates. Write each equation using polar coordinates $(r, \theta)$
$$x=4$$

Aamir Mithaiwala
Aamir Mithaiwala
Numerade Educator
00:24

Problem 76

The letters x and y represent rectangular coordinates. Write each equation using polar coordinates $(r, \theta)$
$$y=-3$$

Aamir Mithaiwala
Aamir Mithaiwala
Numerade Educator
00:58

Problem 77

The letters $r$ and $\theta$ represent polar coordinates. Write each equation using rectangular coordinates $(x, y)$.
$$r=\cos \theta$$

Aamir Mithaiwala
Aamir Mithaiwala
Numerade Educator
01:17

Problem 78

The letters $r$ and $\theta$ represent polar coordinates. Write each equation using rectangular coordinates $(x, y)$.
$$r=\sin \theta+1$$

Aamir Mithaiwala
Aamir Mithaiwala
Numerade Educator
01:00

Problem 79

The letters $r$ and $\theta$ represent polar coordinates. Write each equation using rectangular coordinates $(x, y)$.
$$r^{2}=\cos \theta$$

Aamir Mithaiwala
Aamir Mithaiwala
Numerade Educator
01:11

Problem 80

The letters $r$ and $\theta$ represent polar coordinates. Write each equation using rectangular coordinates $(x, y)$.
$$r=\sin \theta-\cos \theta$$

Aamir Mithaiwala
Aamir Mithaiwala
Numerade Educator
00:45

Problem 81

The letters $r$ and $\theta$ represent polar coordinates. Write each equation using rectangular coordinates $(x, y)$.
$$r=2$$

Aamir Mithaiwala
Aamir Mithaiwala
Numerade Educator
00:22

Problem 82

The letters $r$ and $\theta$ represent polar coordinates. Write each equation using rectangular coordinates $(x, y)$.
$$r=4$$

Aamir Mithaiwala
Aamir Mithaiwala
Numerade Educator
00:55

Problem 83

The letters $r$ and $\theta$ represent polar coordinates. Write each equation using rectangular coordinates $(x, y)$.
$$r=\frac{4}{1-\cos \theta}$$

Aamir Mithaiwala
Aamir Mithaiwala
Numerade Educator
00:55

Problem 84

The letters $r$ and $\theta$ represent polar coordinates. Write each equation using rectangular coordinates $(x, y)$.
$$r=\frac{3}{3-\cos \theta}$$

Aamir Mithaiwala
Aamir Mithaiwala
Numerade Educator
04:18

Problem 85

Chicago In Chicago, the road system is set up like a Cartesian plane, where streets are indicated by the number of blocks they are from Madison Street and State Street.For example, Wrigley Field in Chicago is located at 1060 West Addison, which is 10 blocks west of State Street and 36 blocks north of Madison Street. Treat the intersection of Madison Street and State Street as the origin of a coordinate system, with east being the positive $x-axis.$
(a) Write the location of Wrigley Field using rectangular coordinates.
(b) Write the location of Wrigley Field using polar coordinates. Use the east direction for the polar axis. Express $\theta$ in degrees.
(c) U.S. Cellular Field, home of the White Sox, is located at 35th and Princeton, which is 3 blocks west of State Street and 35 blocks south of Madison. Write the location of U.S. Cellular Field using rectangular coordinates.
(d) Write the location of U.S. Cellular Field using polar coordinates. Use the east direction for the polar axis. Express $\theta$ in degrees.

Katelyn Chen
Katelyn Chen
Numerade Educator
05:53

Problem 86

Show that the formula for the distance $d$ between two points $P_{1}=\left(r_{1}, \theta_{1}\right)$ and $P_{2}=\left(r_{2}, \theta_{2}\right)$ is
$$
d=\sqrt{r_{1}^{2}+r_{2}^{2}-2 r_{1} r_{2} \cos \left(\theta_{2}-\theta_{1}\right)}
$$

Katelyn Chen
Katelyn Chen
Numerade Educator
00:25

Problem 87

In converting from polar coordinates to rectangular coordinates, what formulas will you use?

Katelyn Chen
Katelyn Chen
Numerade Educator
00:46

Problem 88

Explain how to convert from rectangular coordinates to polar coordinates.

Katelyn Chen
Katelyn Chen
Numerade Educator
01:24

Problem 89

Is the street system in your town based on a rectangular coordinate system, a polar coordinate system, or some other system? Explain.

Harmender Singh Yadav
Harmender Singh Yadav
Numerade Educator
04:37

Problem 90

Are based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam.
$$\text { Solve: } \log _{4}(x+3)-\log _{4}(x-1)=2$$

Raushan Kumar
Raushan Kumar
Numerade Educator
02:40

Problem 91

Are based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam.
Use Descartes' Rule of Signs to determine the possible number of positive or negative real zeros for the function. $$f(x)=-2 x^{3}+6 x^{2}-7 x-8$$

Kian Manafi
Kian Manafi
Numerade Educator
02:55

Problem 92

Are based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam.
$$\begin{aligned}
&\text { Find the midpoint of the line segment connecting the }\\
&\text { points }(-3,7) \text { and }\left(\frac{1}{2}, 2\right)
\end{aligned}$$

Jennifer Stoner
Jennifer Stoner
Numerade Educator
01:07

Problem 93

Are based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam.
Given that the point (3,8) is on the graph of $y=f(x)$ what is the corresponding point on the graph of $y=-2 f(x+3)+5 ?$

Katelyn Chen
Katelyn Chen
Numerade Educator