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Trigonometry

Ron Larson, Robert Hostetler

Chapter 6

Topics in Analytic Geometry - all with Video Answers

Educators

Section 1

Lines

00:15

Problem 1

The ____ of a nonhorizontal line is the positive angle $\theta$ (less than $\pi$ ) measured counterclockwise from the $x$-axis to the line.

Amy Jiang
Amy Jiang
Numerade Educator
01:44

Problem 1

In Exercises 1-8, find the slope of the line with inclination $\theta$.

Amy Jiang
Amy Jiang
Numerade Educator
01:59

Problem 2

If a nonvertical line has inclination $\theta$ and slope $m$, then $m=$

Yifan Xu
Yifan Xu
Numerade Educator
01:44

Problem 2

In Exercises 1-8, find the slope of the line with inclination $\theta$.

Amy Jiang
Amy Jiang
Numerade Educator
00:31

Problem 3

If two nonperpendicular lines have slopes $m_{1}$ and $m_{2}$, the angle between the two lines is $\tan \theta=$

Ashly Sunny
Ashly Sunny
Numerade Educator
01:44

Problem 3

In Exercises 1-8, find the slope of the line with inclination $\theta$.

Amy Jiang
Amy Jiang
Numerade Educator
00:42

Problem 4

The distance between the point $\left(x_{1}, y_{1}\right)$ and the line $A x+B y+C=0$ is given by $d=$

Lewis Groves
Lewis Groves
Numerade Educator
01:44

Problem 4

In Exercises 1-8, find the slope of the line with inclination $\theta$.

Amy Jiang
Amy Jiang
Numerade Educator
01:53

Problem 5

$\theta=\frac{\pi}{3}$ radians

Aman Gupta
Aman Gupta
Numerade Educator
01:02

Problem 6

$\theta=\frac{5 \pi}{6}$ radians

Bahar Tehranipoor
Bahar Tehranipoor
Numerade Educator
01:02

Problem 7

$\theta=1.27$ radians

Bahar Tehranipoor
Bahar Tehranipoor
Numerade Educator
01:02

Problem 8

$\theta=2.88$ radians

Bahar Tehranipoor
Bahar Tehranipoor
Numerade Educator
01:44

Problem 9

In Exercises 9-14, find the inclination $\theta$ (in radians and degrees) of the line with a slope of $m$.
$m=-1$

Amy Jiang
Amy Jiang
Numerade Educator
00:33

Problem 10

$m=-2$

Nick Johnson
Nick Johnson
Numerade Educator
00:21

Problem 11

$m=1$

Nick Johnson
Nick Johnson
Numerade Educator
00:33

Problem 12

$m=2$

Nick Johnson
Nick Johnson
Numerade Educator
01:00

Problem 13

$m=\frac{3}{4}$

Ramzi Deek
Ramzi Deek
Numerade Educator
00:20

Problem 14

$m=-\frac{5}{2}$

Nick Johnson
Nick Johnson
Numerade Educator
00:40

Problem 15

In Exercises 15-18, find the inclination $\theta$ (in radians and degrees) of the line passing through the points.
$(6,1),(10,8)$

Amy Jiang
Amy Jiang
Numerade Educator
00:08

Problem 16

$(12,8),(-4,-3)$

James Kiss
James Kiss
Numerade Educator
00:20

Problem 17

$(-2,20),(10,0)$

James Kiss
James Kiss
Numerade Educator
00:15

Problem 18

$(0,100),(50,0)$

James Kiss
James Kiss
Numerade Educator
00:40

Problem 19

In Exercises 19-22, find the inclination $\theta$ (in radians and degrees) of the line.
$6 x-2 y+8=0$

Amy Jiang
Amy Jiang
Numerade Educator
01:25

Problem 20

$4 x+5 y-9=0$

Dwijendra Rao
Dwijendra Rao
Numerade Educator
00:35

Problem 21

$5 x+3 y=0$

Nick Johnson
Nick Johnson
Numerade Educator
00:56

Problem 22

$x-y-10=0$

Rylie Howey
Rylie Howey
Numerade Educator
01:17

Problem 23

In Exercises 23-32, find the angle $\theta$ (in radians and degrees) between the lines.
$$
\begin{array}{r}
3 x+y=3 \\
x-y=2
\end{array}
$$

Ashly Sunny
Ashly Sunny
Numerade Educator
00:33

Problem 24

$$
\begin{aligned}
&x+3 y=2 \\
&x-2 y=-3
\end{aligned}
$$

Nick Johnson
Nick Johnson
Numerade Educator
02:51

Problem 25

$$
\begin{array}{r}
x-y=0 \\
3 x-2 y=-1
\end{array}
$$

Gregory Higby
Gregory Higby
Numerade Educator
02:43

Problem 26

$$
\begin{aligned}
&2 x-y=2 \\
&4 x+3 y=24
\end{aligned}
$$

AG
Ankit Gupta
Numerade Educator
02:00

Problem 27

$$
\begin{array}{r}
x-2 y=7 \\
6 x+2 y=5
\end{array}
$$

Amanda Tricario
Amanda Tricario
Numerade Educator
01:29

Problem 28

$$
\begin{aligned}
&5 x+2 y=16 \\
&3 x-5 y=-1
\end{aligned}
$$

Rylie Howey
Rylie Howey
Numerade Educator
03:14

Problem 29

$$
\begin{aligned}
&x+2 y=8 \\
&x-2 y=2
\end{aligned}
$$

Sonam Khatri
Sonam Khatri
Numerade Educator
01:36

Problem 30

$$
\begin{aligned}
&3 x-5 y=3 \\
&3 x+5 y=12
\end{aligned}
$$

Rylie Howey
Rylie Howey
Numerade Educator
05:40

Problem 31

$$
\begin{aligned}
&0.05 x-0.03 y=0.21 \\
&0.07 x+0.02 y=0.16
\end{aligned}
$$

Aman Gupta
Aman Gupta
Numerade Educator
05:40

Problem 32

$$
\begin{aligned}
&0.02 x-0.05 y=-0.19 \\
&0.03 x+0.04 y=0.52
\end{aligned}
$$

Aman Gupta
Aman Gupta
Numerade Educator
03:32

Problem 33

Angle Measurement In Exercises 33-36, find the slope of each side of the triangle and use the slopes to find the measures of the interior angles.

Kimberly Waterbury
Kimberly Waterbury
Numerade Educator
03:32

Problem 34

Angle Measurement In Exercises 33-36, find the slope of each side of the triangle and use the slopes to find the measures of the interior angles.

Kimberly Waterbury
Kimberly Waterbury
Numerade Educator
03:32

Problem 35

Angle Measurement In Exercises 33-36, find the slope of each side of the triangle and use the slopes to find the measures of the interior angles.

Kimberly Waterbury
Kimberly Waterbury
Numerade Educator
03:32

Problem 36

Angle Measurement In Exercises 33-36, find the slope of each side of the triangle and use the slopes to find the measures of the interior angles.

Kimberly Waterbury
Kimberly Waterbury
Numerade Educator
01:59

Problem 37

In Exercises 37-44, find the distance between the point and the line.
Point
Line
$(0,0) \quad 4 x+3 y=0$

Dwijendra Rao
Dwijendra Rao
Numerade Educator
02:05

Problem 38

$(0,0) \quad 2 x-y=4$

Rylie Howey
Rylie Howey
Numerade Educator
01:54

Problem 39

$(2,3) \quad 4 x+3 y=10$

Heather Zimmers
Heather Zimmers
Numerade Educator
00:37

Problem 40

$(-2,1) \quad x-y=2$

Nick Johnson
Nick Johnson
Numerade Educator
02:53

Problem 41

$(6,2) \quad x+1=0$

Sherrie Fenner
Sherrie Fenner
Numerade Educator
01:41

Problem 42

$(10,8) \quad y-4=0$

Adriano Chikande
Adriano Chikande
Numerade Educator
03:41

Problem 43

$(0,8) \quad 6 x-y=0$

Ian Grigsby
Ian Grigsby
Numerade Educator
00:52

Problem 44

$(4,2) \quad x-y=20$

Rylie Howey
Rylie Howey
Numerade Educator
04:11

Problem 45

In Exercises 45-48, the points represent the vertices of a triangle. (a) Draw triangle $A B C$ in the coordinate plane, (b) find the altitude from vertex $B$ of the triangle to side $A C$, and (c) find the area of the triangle.
$A=(0,0), B=(1,4), C=(4,0)$

Amy Jiang
Amy Jiang
Numerade Educator
03:09

Problem 46

$A=(0,0), B=(4,5), C=(5,-2)$

Shahab Ullah
Shahab Ullah
Numerade Educator
01:54

Problem 47

$A=\left(-\frac{1}{2}, \frac{1}{2}\right), B=(2,3), C=\left(\frac{5}{2}, 0\right)$

Harmender Singh Yadav
Harmender Singh Yadav
Numerade Educator
01:05

Problem 48

$A=(-4,-5), B=(3,10), C=(6,12)$

Hast Aggarwal
Hast Aggarwal
Numerade Educator
04:32

Problem 49

In Exercises 49 and 50, find the distance between the parallel lines.
$$
\begin{aligned}
&x+y=1 \\
&x+y=5
\end{aligned}
$$

Adam Cornes
Adam Cornes
Numerade Educator
01:54

Problem 50

$$
\begin{aligned}
&3 x-4 y=1 \\
&3 x-4 y=10
\end{aligned}
$$

Heather Zimmers
Heather Zimmers
Numerade Educator
01:51

Problem 51

A straight road rises with an inclination of $0.10$ radian from the horizontal (see figure). Find the slope of the road and the change in elevation over a two-mile stretch of the road.

Tani Iqbal
Tani Iqbal
Numerade Educator
02:06

Problem 52

A straight road rises with an inclination of $0.20$ radian from the horizontal. Find the slope of the road and the change in elevation over a one-mile stretch of the road.

Ashly Sunny
Ashly Sunny
Numerade Educator
01:28

Problem 53

A roof has a rise of 3 feet for every horizontal change of 5 feet (see figure). Find the inclination of the roof.

Suman Saurav Thakur
Suman Saurav Thakur
Numerade Educator
02:54

Problem 54

A moving conveyor is built so that it rises 1 meter for each 3 meters of horizontal travel.
(a) Draw a diagram that gives a visual representation of the problem.
(b) Find the inclination of the conveyor.
(c) The conveyor runs between two floors in a factory. The distance between the floors is 5 meters. Find the length of the conveyor.

Kian Manafi
Kian Manafi
Numerade Educator
07:37

Problem 55

Find the angles $\alpha$ and $\beta$ shown in the drawing of the roof truss.

Adam Cornes
Adam Cornes
Numerade Educator
02:05

Problem 56

The Johnstown Inclined Plane in Johnstown, Pennsylvania is an inclined railway that was designed to carry people to the hilltop community of Westmont. It also proved useful in carrying people and vehicles to safety during severe floods. The railway is $896.5$ feet long with a $70.9 \%$ uphill grade (see figure).
(a) Find the inclination $\theta$ of the railway.
(b) Find the change in elevation from the base to the top of the railway.
(c) Using the origin of a rectangular coordinate system as the base of the inclined plane, find the equation of the line that models the railway track.
(d) Sketch a graph of the equation you found in part (c).

AG
Ankit Gupta
Numerade Educator
00:30

Problem 57

True or False? In Exercises 57 and 58, determine whether the statement is true or false. Justify your answer.
A line that has an inclination greater than $\pi / 2$ radians has a negative slope.

Amy Jiang
Amy Jiang
Numerade Educator
01:03

Problem 58

To find the angle between two lines whose angles of inclination $\theta_{1}$ and $\theta_{2}$ are known, substitute $\theta_{1}$ and $\theta_{2}$ for $m_{1}$ and $m_{2}$, respectively, in the formula for the angle between two lines.

Carson Merrill
Carson Merrill
Numerade Educator
05:44

Problem 59

Consider a line with slope $m$ and $y$-intercept $(0,4)$.
(a) Write the distance $d$ between the origin and the line as a function of $m$.
(b) Graph the function in part (a).
(c) Find the slope that yields the maximum distance between the origin and the line.
(d) Find the asymptote of the graph in part (b) and interpret its meaning in the context of the problem.

Olivier Anderson
Olivier Anderson
Numerade Educator
08:36

Problem 60

Consider a line with slope $m$ and $y$-intercept $(0,4)$.
(a) Write the distance $d$ between the point $(3,1)$ and the line as a function of $m$.
(b) Graph the function in part (a).
(c) Find the slope that yields the maximum distance between the point and the line.
(d) Is it possible for the distance to be 0? If so, what is the slope of the line that yields a distance of 0 ?
(e) Find the asymptote of the graph in part (b) and interpret its meaning in the context of the problem.

Olivier Anderson
Olivier Anderson
Numerade Educator
01:35

Problem 61

In Exercises 61-66, find all $x$-intercepts and $y$-intercepts of the graph of the quadratic function.
$f(x)=(x-7)^{2}$

Deboney Harris
Deboney Harris
Numerade Educator
View

Problem 62

$f(x)=(x+9)^{2}$

Ashwin Narayan
Ashwin Narayan
Numerade Educator
02:19

Problem 63

$f(x)=(x-5)^{2}-5$

Taylor Shimono
Taylor Shimono
Numerade Educator
01:38

Problem 64

$f(x)=(x+11)^{2}+12$

Taylor Shimono
Taylor Shimono
Numerade Educator
08:17

Problem 65

$f(x)=x^{2}-7 x-1$

Bobby Barnes
Bobby Barnes
University of North Texas
View

Problem 66

$f(x)=x^{2}+9 x-22$

Ashwin Narayan
Ashwin Narayan
Numerade Educator
01:14

Problem 67

In Exercises 67-72, write the quadratic function in standard form by completing the square. Identify the vertex of the function.
$f(x)=3 x^{2}+2 x-16$

Erika Bustos
Erika Bustos
Numerade Educator
01:38

Problem 68

$f(x)=2 x^{2}-x-21$

James Kiss
James Kiss
Numerade Educator
06:27

Problem 69

$f(x)=5 x^{2}+34 x-7$

Bobby Barnes
Bobby Barnes
University of North Texas
01:42

Problem 70

$f(x)=-x^{2}-8 x-15$

Taylor Shimono
Taylor Shimono
Numerade Educator
01:07

Problem 71

$f(x)=6 x^{2}-x-12$

Carson Merrill
Carson Merrill
Numerade Educator
03:17

Problem 72

$f(x)=-8 x^{2}-34 x-21$

Nick Johnson
Nick Johnson
Numerade Educator
03:32

Problem 73

In Exercises 73-76, graph the quadratic function.
$f(x)=(x-4)^{2}+3$

Suman Saurav Thakur
Suman Saurav Thakur
Numerade Educator
01:36

Problem 74

$f(x)=6-(x+1)^{2}$

Gregory Higby
Gregory Higby
Numerade Educator
00:38

Problem 75

$g(x)=2 x^{2}-3 x+1$

James Kiss
James Kiss
Numerade Educator
02:37

Problem 76

$g(x)=-x^{2}+6 x-8$

Gregory Higby
Gregory Higby
Numerade Educator