Modeling Wavy Rock Formations

TEAChER’SEDITION
Precalculus with Trigonometry Course Sampler 69
y
x
Stresses in the earth compress rock
formations and cause them to buckle
into sinusoidal shapes. It is important for
geologists to be able to predict the depth
of aof aof rock formation at a given point. Such
information can be very usefry usefry ulusefulusef for structural
engineers as well. In this chapter you’ll
learn about the circular functions, which
are closely related to the trigonometric
functions. Geologists and engineers use
these functions as mathematical models to
perforrforrf m calculations for such wavych wavych wa rock
formations.
ApplicationsofTrigonometric
andCircularFunctions
281
CHAPTER OBJECTIVES
yyy
x

TEAChER’SEDITION
70 Precalculus with Trigonometry Course Sampler
281A Chapter 6 Interleaf: Applications of Trigonometric and Circular Functions
Overview
Using This Chapter
Teaching Resources
Explorations
y x
yyy x
Blackline Masters
Supplementary Problems
Assessment Resources
Technology Resources
Dynamic Precalculus Explorations
x x x
Sketchpad Presentation Sketches
Activities
Applications of Trigonometric
and Circular Functions
Applications of Trigonometric
and Circular Functions
Chapter 6

TEAChER’SEDITION
Chapter 6 Interleaf 281B
Standard Schedule Pacing Guide
Block Schedule Pacing Guide
Day Section Suggested Assignment
y, x
Day Section Suggested Assignment
3
y, x

TEAChER’SEDITION
Chapter 6:
y y
y
xxx
: y x
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GRAPHICALLY
ALGEBRAICALLY
NUMERICALLY
x yx y
VERBALLY
Mathematical Overview
x
y
282 Chapter 6: Applications of Trigonometric and Circular Functions
CAS Suggestions
fff xxx x
y x
y x
Section 6-1
PLANNING
Class Time
__
Homework Assignment
Teaching Resources
TEACHING
Important Terms and Concepts
Section Notes
Exploration 6-1a
Exploration Notes
Exploration 6-1a

TEAChER’SEDITION
Section 6-1:
Sinusoids: Amplitude, Period,
and Cycles
Figure 6-1a shows a dilated and
translated sinusoid and some of its
graphical features. In this section you
will learn how these features relate to
transforsforsf mations you’ve already learned.
FiguFiguFi re 6-1a
Learn the meanings of amplitudeearn the meanings of amplitudeearn the meanings of , perioamplitude, perioamplitude d, phase displacement, and cyclecyclecy of a
sinusoidal graph.
Sinuso
and Cy
6-1
Learn
sinus
Objective
1. Sketch one cycle of the grof the grof aph of the paof the paof rent
sinusoid y cos , starting at 0°. What is
the amplitude of this grof this grof aph?
2. Plot the graph of the transfof the transfof orthe transforthe transf med cosine
function y 5 cos . What is the amplitude of
this graph? What is the relationship between the
amplitude and the vertical dilatirtical dilatirtical dila on of a sinof a sinof usoid?
3. What is the period of the transfof the transfof orthe transforthe transf med function
in Problem 2? What is the period of the paof the paof rent
function y cos ?
4. Plot the graph of y cos 3 . What is the period
of this transfof this transfof orthis transforthis transf med function graph? How is the
3 related to the transford to the transford to the transf mation? How could you
calculate the period using the 3?
5. Plot the graph of y cos( 60°). What
transformation is caused by the 60°?by the 60°?by
6. The ( 60°) in Problem 5 is called the
arguarguar ment of the cosine.of the cosine.of The phase displacement
is the value of that makes the argument equal
zero. What is the phase displacement for this
function? How is the phase displacement related
to the horizontal translation?
7. Plot the graph of y 6 cos . What
transformation is caused by the 6?by the 6?by
8. The sinusoidal axis runs along the middle of theof theof
graph of a sinof a sinof usoid. It is the dashed centerline in
Figure 6-1a. What transformation of the function
y cos x does the locax does the locax tion of the sinof the sinof usoidal axis
indicate?
9. What are the amplitude, period, phase
displacement, and sinusoidal axis location of theof theof
graph of y 6 5 cos 3( 60°)? Check by
plotting on your grapher.
1. Sketch one cycle of the graph of the parent 5. Plot the graph of y cos( 60°). What
Exploratory Problem Set 6-1
y
One cycle
Sinusoidal axis
Amplitude
Period
Phase displacement
(horizontal translation)
plitude
283Section 6-1: Sinusoids: Amplitude, Period, and Cycles
PROBLEM NOTES
Problems 2–5, 7, and 9 can beProblems 2–5, 7, and 9 can beProblems 2–5, 7, and 9
solved using the approaches described in
the CAS Suggestions.
2. Amplitude 5
2
2
360°
y
180°
The absolute value of the vertical dilation
of a sinusoid equals the amplitude.
3. 360 for both functions
4. 120
2
2
360°
y
The 3 is the reciprocal of the horizontal
dilation. The period equals 360 divided
by 3.
5.
Horizontal translation by 60
6. Phase displacement 60 . The phase
displacement equals the horizontal
translation.
7.
Vertical translation by 6
8. The vertical translation corresponds
to the sinusoidal axis.
9. Amplitude 5; Period 120 ; Phase
displacement 60 ; Sinusoidal axis 6
360°
y
8
2
2
360°
y
2
2
360°
y
1. Amplitude 1
2
2
360°
y

TEAChER’SEDITION
Chapter 6:
General Sinusoidal Graphs
period, amplitude, cycle, phase
displacement,ment,ment sinusoidal axis.
Genera6-2
Objective
y
x
1. y
y
1
EXPLORATION 6-2: Periodic Daily Temperatures
continued
Instructor’s Resource Book
1. y
2. y
3. y
Section 6-2
PLANNING
Class Time
Homework Assignment
Day 1
Day 2:
Teaching Resources
TEACHING
Exploration Notes
Exploration 6-2

TeacheR’sEdition
285Section 6-2: General Sinusoidal Graphs
frequency.
DEFINITION: General Sinusoidal Equation
y C A B D y C A B D
A A
B
C
D
2.
12° 24°
y
18°6°
−1
3.
10
20
−10
−20
12° 24°
y
7°
4.
5.
6.
EXPLORATION, continued
Section Notes
Exploration 6-2a
Exploration 6-2b
4. y
5. y
y
6.

Teacher’sEdition
B. B
B
PROPERTIES: Period and Frequency of a Sinusoid
y C A B D y C A B D
____
B
B____
y
Figure 6-2a
upper bound lower bound,
critical points
y
Figure 6-2b
Section Notes (continued)
Concavity, point
of inflection, upper bound, lower
bound
frequency,
A,
B, C, D
y C A B D
y C A B D
B.
B ____
period
period____
y
every

TEAChER’SEDITION
Section 6-2:
y
y
θ
Figure 6-2c
y
Figure 6-2d
y
Figure 6-2e
EXAMPLE 1
SOLUTION
287
y
f
f
f
x
x
of degrees.
degrees
Domain Range
Example 1 y
Example 2 y
Examples 3 and 4 y
Section 6-2: General Sinusoidal Graphs

Teacher’sEdition
y
Figure 6-2f
B
y
y
Figure 6-2h
EXAMPLE 2
y
Figure 6-2g
after
Astronomy
Mathematics: An Introduction
to Its Spirit and Use
National Geographic Differentiating Instruction
www.keypress.com/
keyonline,
argument
bound frequency
u v

TeacheR’sEdition
y
A
B
y
y
Figure 6-2i
y
y
SOLUTION
EXAMPLE 3
SOLUTION
289
Refer to the Problem Notes for
Problems 5–8 and Problems 9–18.
These problems will help prepare
students for Section 6-7, which will
present a challenge for ELL students
because of the complex language.
Problem 28 will challenge ELL students
to develop their English skills.
Additional Exploration Notes
Exploration 6-2a gives students practice
in graphing particular cosine and sine
equations; identifying period, amplitude,
and other characteristics; writing an
equation from a given graph; checking
their work using a grapher; and finding
a y-value given an angle value. Assign
this exploration as a group activity
after you have presented and discussed
Examples 1 and 2. This gives students
supervised practice with the ideas
presented in the examples before they
leave class. Students may need help with
Problem 5, which asks them to find y
when 35°. Allow 20–30 minutes for
the exploration.
Exploration 6-2b can be assigned as a
quiz or as extra homework practice. It
can also be assigned later in the year as
a review of sinusoids. Problem 1 of the
exploration asks students to sketch two
cycles of a given sinusoid. Problems 2
and 3 ask them to write two equations—
one using sine and one using cosine—for
the same sinusoid and then to check
their equations using their graphers.
Problems 5 and 6 give students practice
writing equations based on a half-cycle
or quarter-cycle of a sinusoid. Allow
20 minutes for the entire exploration or
8–10 minutes for just Problems 5 and 6.
to trace with their fingers to reinforce
their understanding and to enable you
to verify that the understanding does
(or doesn’t) exist. These graphs can also
help reinforce the meanings of sinusoid,
periodic, and displacement, and clarify
translation of one period, pre-image and
image graphs, and negative angles.
Have students use their paper graphs
while you go over the examples.
Have students write the general
sinusoidal equation definition in their
journals both formally and in their
own words.
Make sure the distinction on page 290
between a particular equation and the
particular equation is clear.
Consider using the Reading Analysis
as an in-class discussion. Alternatively,
allow ELL students to do it in pairs.

TEAChER’SEDITION
Chapter 6:
y
FiguFiguFi re 6-2j-2j-2
B
y
a the
a.
b.
c.
d.
y
FiguFiguFi re 6-2k
EXAMPLE 4
Technology Notes
Exploration 6-2
CAS Suggestions
Note:
absolutely

TEAChER’SEDITION
Section 6-2:
a.
y
b.
goinggoinggoin up.
y y
c.
low
y
d.
down.
y
a.SOLUTION
ReadingAnalysis
period, frequency,equency,equenc cycle
uv
Quick Review
Q1.
Q2.
y
FiguFiguFi re 6-2l
Q3.
Q4.
Q5.
Q6.
Q7.Q7.Q7
Q8.
5min
Reading Analysis y
Problem Set 6-2
291
PROBLEM NOTES
Q1.
Q2.
Q3.
Q4.
Q5. y x
Q6.
__
____
Q7.
Q8.

TEAChER’SEDITION
Chapter 6:
Q9.
Q10. : x
1. y
2. y
3. y
4. y
a.
b.
c. y
..
5.
y
6.
y
7.
y
8.
y
9.
10.
y
y
Problem Notes (continued)
Q9. tan 1 11___
9
50.7105...
Q10. 9x2 30x 25
Problems 1–4 should all be assigned
so students get sufficient practice in
graphing transformed sinusoids.
1. Amplitude 4; Period 120 ;
Phase displacement 10 ;
Sinusoidal axis 7
360°
y
4
Sinusoidal axis 3
360°
y
2
2
Sinusoidal axis 10
360°
y
20
Sinusoidal axis 8
5
10
100° 200°
15
20
y
The Solve command could be
used to determine the coefficient of in
Problems 5–22.
Problems 5–8 are important not only
because students must create equations for
given graphs but also because they must
find y-values for given angle values. This
is an excellent preview of Section 6-7, in
which students use sinusoidal equations to
predict results in real-world problems. Be
sure to discuss finding the y-values during
the class discussion of the homework. A
blackline master for these problems is
available in the Instructor’s Resource Book.
5a. y 9 6cos 2( 20 )
5b. Amplitude 6; Period 180 ;
Frequency 1___
180 cycle/deg;
Sinusoidal axis 9
5c. y 10.0418... at 60 ;
y 8.7906... at 1234

TeacheR’sEdition
11.
12.
13.
14.
15.
16.
17.
18.
19.
y
20.
y
21.
y
y
22.
y
y
y
y
r
y
y
y
y
y
293
8a. y 30 10sin 18( 2 )
Frequency 1__
20 cycle/deg;
Sinusoidal axis 30
8c. y 30 at 8 ;
y 35.8778... at 1776
Problems 9–18 provide practice in
writing equations from graphs that show
complete cycles or parts of cycles.
This is another critical skill required
to solve the real-world problems in
Section 6-7. A blackline master for these
problems is available in the Instructor’s
Resource Book.
9. y 1.45 1.11sin 10( 16 )
10. y 30 20cos 360( 0.3 )
11. y 1.7cos( 30 )
12. y 5000cos 45( 1 )
Problems 13 and 14 use different variables
to name the horizontal and vertical axes.
When you discuss homework, make sure
students have used the correct variables
and not just the usual variables y and .
13. r 7 cos 3
14. y 0.03 sin 0.9
15. y 35 15sin 90
16. y 7 3sin 4.5
17. y 4 9sin 9__
13( 60 )
18. y 60 100sin 45
Problems 19 and 20 require students to
find y-values corresponding to given
angle values.
19. y 12.4151... at 300;
y 2.0579..., 1.9420... below the
sinusoidal axis at 5678
20. y 32.3879... at 2.5 ; y 60,
on the sinusoidal axis, at 328
Problems 21 and 22 require students to
write equations for sinusoids based on
verbal descriptions.
6a. y 8 10cos 9( 4 )
Frequency 1__
40 cycle/deg;
Sinusoidal axis 8
6c. y 2.1221... at 10 ;
y 6.4356... at 453
7a. y 3 5cos 3( 10 )
Frequency 1___
120 cycle/deg;
Sinusoidal axis 3
7c. y 8 at 70 ;
y 1.9931... at 491
See page 1002 for answers to
Problems 21 and 22.

TEAChER’SEDITION
Chapter 6:
a.
b.
c.
d.
23.
24.
25.
a.
b.
26.
a.
b.
c.
27.
a.
b.
c.
28.
Problems 23 and 24
Instructor’s
Resource Book
Problems 23 and 24
Problem 25
25a.
25b.
_
Problem 26
inflection point, concave up, concave
down
Problem 27
Problem 28
28.
Additional CAS Problems
1.
y x
y xxx
x xxx
x
2. xxx
x xxx
x?
3. Solve(y = f(x), y) | x = 0
f xxx
4. Solve(y = f(x), x) | y = 0
f xxx

TEAChER’SEDITION
Section 6-3:
Graphs of Tangent, Cotangent, Secant,
and Cosecant Functions
vertical asymptotes
discontinuous
y y
y y
y y
y y
FiguFiguFi re 6-3a
y
y y
FiguFiguFi re 6-3b
Graph
and Co
6-3
Objective
295Section 6-3: Graphs of Tangent, Cotangent, Secant, and Cosecant Functions
Section 6-3
PLANNING
Class Time
Homework Assignment
Teaching Resources
TEACHING
Section Notes

TEAChER’SEDITION
Chapter 6:
y .
.
y
FiguFiguFi re 6-3c
u vvv uv rrr
u vvv
v
u
rrr
v/rv/rv/___
u/u/ru/ru/
. quotient properties
PROPERTIES: Quotient Properties for Tangent and Cotangent
y y .
y .
EXAMPLE 1
SOLUTION
y
FiguFiguFi re 6-3d
EXAMPLE 2
y
y
y
y y
y
y y
y
MODE DOT
CONNECTED
____ y

TeacheR’sEdition
297
y y
Figure 6-3f Figure 6-3g
SOLUTION
y
Section 6-3: Graphs of Tangent, Cotagent, Secant, and Cosecant Functions
ReadingAnalysis
Quick Review
y
Q1.
Q2.
Q3.
Q4.
y
Q5.
Q6. y
Q7. y
Q8. y x
Q9. y x
Q10.
A. B.
C. D.
E.
1. Secant Function Problem
a.
y .
y .
b.
c.
d.
5min
Reading Analysis Q8 y x
Problem Set 6-3
Figure 6-3e
297
Differentiating Instruction
Problem 7
Problem 9
y
(location of 1st
asymptote) (distance between
asymptotes) n, n .
Exploration Notes
Exploration 6-3a
Exploration 6-3b
Function Period Domain Range
y tan
n, n
y
y cot
n, n
y
y sec
n, n
y y
y csc
n, n
y y
Section 6-3: Graphs of Tangent, Cotangent, Secant, and Cosecant Functions

TEAChER’SEDITION
Chapter 6:
2. Tangenngenn t Function Problem
a.
y
y
b.
y
c.
y ..
d.
e.
3. Quotient Prnt Prnt operty for Tangenngenn t Problem:
fff xxx
fff xxx
fff xxx ffff xxx ffff xxx
4. Quotient Prnt Prnt operty for Cotangent Problem:
fff xxx
fff xxx
fff xxx ffff xxx ffff xxx
5.
y y ..
6.
y y
7.
y y
8.
9. y
10. y
11. y
12. y
13. y
14. y
15. Rotating LighLighLi thouse Beacon Problem:
D
FiguFiguFi re 6-3h
Technology Notes
Problem 16 asks students toProblem 16 asks students toProblem 16
explore the behaviors of the six
trigonometric functions as the
angle of measurement varies. They
may create their own sketches
using Sketchpad, or they may
use the Variation of Tangent and
Secant exploration at
www.keymath.com/precalc.
Presentation Sketch:
Trig Tracers Present.gsp at
www.keypress.com/keyonline
is a presentation sketch that
traces out the graphs of the six
trigonometric functions as a point
is animated on a unit circle.
Exploration 6-3a: Tangent and
Secant Graphs has students
discover what the graphs of tangent
and secant look like and how
they relate to the graphs of sine
and cosine. The better resolution
available in Fathom may be useful
for the parts where they are asked
to use a grapher. Sketchpad could
also be employed here.
PROBLEM NOTES
Supplementary problems for this section
are available at www.keypress.com/
keyonline.
Problems 1–4 reinforce the work done in
Exploration 6-3a.
In Problem 2b, the -intercepts
and vertical asymptotes can be
found on a CAS by solving equations
involving the corresponding portions
of the tangent fraction. Notice that the
vertical asymptote locations are written
in an uncommon, but correct, form.
Recognizing equivalent forms is a critical
mathematical skill, especially when using
a CAS.
2a., 2c.
90° 450°
y
1
2b. Asymptotes occur where cos 0 at
90 180 n; -intercepts occur where
sin 0 at 180 n.

TeacheR’sEdition
299Section 6-3: Graphs of Tangent, Cotangent, Secant, and Cosecant Functions
D
a. D .
D
b.
c.
D D
d.
16. Variation of Tangent and Secant
Problem:
uv
O, ,
P.
P,
u v B.
P
u C,
P v D.
v
u
O
P
A
B
D
C
P
Figure 6-3i
a.
PA
PB
PC
PD
OA
OB
b. v
complement
.
co
complement .
c.
Variation
of Tangent and Secant
299
8.
9. ____
n.
y
10. ____
n
Problems 11–14
Problem 15
L
2c.
2d.
2e. n
Problems 5–10
learn
5.
6.
7. n
n n
n
n__________
n
_____
____
n
n__________
n
_____
____
Section 6-3: Graphs of Tangent, Cotangent, Secant, and Cosecant Functions

TEAChER’SEDITION
Chapter 6:
Radian Measure of Angles
radian
Radian6-4
Objective
rrr
uv
u
v
u
r r
1.
u
r
2.
u
3.
4.
5.
1
EXPLORATION 6-4: Introduction to Radians
Section 6-4
PLANNING
Class Time
Homework Assignment
Teaching Resources
TEACHING
Exploration Notes
Exploration 6-4
Exploration 6-4
1.
0.5 r
1 r 1 r 1 r
Index Card
r
r

TeacheR’sEdition
301Section 6-4: Radian Measure of Angles
uv
r
v
u
v
u
r
Figure 6-4a
subtends
x
x x
a
r
a
r
radian
measure
Excerpt from an old
Babylonian cuneiform text
x
r
r x
Figure 6-4c
r
r
a
a
Figure 6-4b
301
Section Notes
Section 6-4 introduces radian measures
for angles. Measuring angles in radians
makes it possible to apply trigonometric
functions to real-life units of measure,
such as time, that can be represented
on the number line. Try to do at least one
of the explorations for Section 6-4 so that
students will understand that a radian
is really the arc length of a circle with a
radius of one unit.
The explorations also help students
understand the relationship between
radians and degrees. If they do not
complete one of these activities, students
may simply memorize how to convert
from degrees to radians without
understanding how the units are related.
Explain that degrees and radians are two
units for measuring angles, just as feet
and meters are two units for measuring
length. An angle measuring 30 radians is
not the same size as an angle measuring
30 degrees (just as a length of 30 meters
is not equal to a length of 30 feet) because
the units are different.
The relationship between radians
and degrees can be used to convert
measurements from one unit to the
other. Because radians is equivalent
to 180 degrees, the conversion factor
is radians________
180 degrees
, a form of 1. Students should
realize that multiplying a number of
degrees by this factor causes the degrees
to cancel out, leaving a number of
radians.
30 degrees_________
1
radians__________
180 degrees
1__
6
radians
Similarly, multiplying a number of
radians by
180 degrees________
radians
, another form of 1,
lets the radians cancel, leaving a number
of degrees. If the units don’t cancel, the
student has the conversion factor upside
down.
2. Arc has length 2.8r, meaning that is
2.8 radians.
3. A full revolution is 2 radians because
the circumference of a circle is 2 r. An
angle of 180° is half of 2 , or radians. An
angle of 90° is 1_
4 of 2 , or _
2 radians.
4. One radian is 1__
2 of a full circle, so its
degree measure is 360°___
2 57.2957...°
5. Answers will vary.
Section 6-4: Radian Measure of Angles

Teacher’sEdition
DEFINITION: Radian Measure of an Angle
radian measure
arc length_________
radius
For the work that follows, it is important to distinguish between the name
of the angle and the measure of that angle. Measures of angle will be written
this way:
is the name of the angle.
m°( ) is the degree measure of angle .
mR
( ) is the radian measure of angle .
Because the circumference of a circle is 2 r and because r for the unit circle
is 1, the wrapped number line in Figure 6-4a divides the circle into 2 units
(a little more than six parts). So there are 2 radians in a complete revolution.
There are also 360° in a complete revolution. You can convert degrees to radians,
or the other way around, by setting up these proportions:
mR
( )_____
m°( )
2____
360°
____
180°
or
m°( )_____
mR
( )
360°____
2
180°____
Solving for mR
( ) and m°( ), respectively, gives
mR
( ) ____
180°
m°( ) and m°( ) 180°____ mR
( )
These equations lead to a procedure for accomplishing the objective of this section.
PROCEDURE: Radian–Degree Conversion
To find the radian measure of , multiply the degree measure by ___
180
.
To find the degree measure of , multiply the radian measure by 180___.
Convert 135° to radians.
In order to keep the units straight, write
each quantity as a fraction with the proper
units. If you have done the work correctly,
certain units will cancel, leaving the proper
units for the answer.
mR
( )
135 degrees__________
1
radians__________
180 degrees
3__
4
2.3561... radians
Notes:
If the exact value is called for, leave the answer as 3_
4 . If not, you have the choice
of writing the answer as a multiple of or converting to a decimal.
The procedure for canceling units used in Example 1 is called dimensional
analysis. You will use this procedure throughout your study of mathematics.
CoEXAMPLE 1
In order
each qua
ISOLUTION x 135 x=
135°
180° 180
__ __
__

TeacheR’sEdition
Radian Measures of Some Special Angles
EXAMPLE 2
SOLUTION
EXAMPLE 3
SOLUTION
EXAMPLE 4
SOLUTION
radians
circumference perimeter
Problems 1
and 2

TEAChER’SEDITION
Chapter 6:
ReadingAnalysis Quick Review
Q1. y
Q2. y
Q3.
y
Q4.
Reading Analysis Quick Review
Problem Set 6-4
v
u
v
u
FiguFiguFi re 6-4d FiguFiguFi re 6-4e
PROPERTY: Radian Measures of Some Special Angles
Degrees
Radians
Revolutions
EXAMPLE 5
SOLUTION
30°
60°
1
2
3
1
cos θ
hypotenuse
adjacent
sec θ = =
Exploration 6-4a
Technology Notes
__
CAS Activity 6-4a:
PROBLEM NOTES
Q1.
Q2.
y
90°
1
y
90°
1
Q3.
Q4.

TeacheR’sEdition
Q5.
Q6. f
g x f x
Q7. f
h x f x
Q8.
Q9.
Q10.
A. B.
C. D.
E.
1. Wrapping Function Problem:
uv
x xy
x
u, v x
a. x
b.
c.
v
x
u
Figure 6-4f
2. Arc Length and Angle Problem:
r
r
r
Figure 6-4g
a.
b.
c. r
d. quickly a
r
305
Q5.
__
Q6. y-
Q7. x
Q8. f x ax bx c, a
Q9. Q10.
Problem 1
Exploration 6-4
Problem 1 Instructor’s
Resource Book
1a.
1b.
1c.
Problem 2
2a.
2b. r
r
2c. r
2d. a r
1
2
3
12
3 u
v
r 1
1
2
3
12
3 u
v
r 1
Section 6-4: Radian Measure of Angles

TEAChER’SEDITION
Chapter 6:
exacttt
3. 4.
5. 6.
7. 8.
9. 10.
11. 12.
13. 14.
15. 16.
17. 18.
19. 20.
21. 22.
23. 24.
25. 26.
27. 28.
29. 30.
31. 32.
33. 34.
5
1
x
35. 36.
37. 38.
exacttt
39. 40.
41. 42.
43. 44.
exacttt
45. 46.
47. 48.
49. y
50.
xxx
51. 52.
53. 54.
y
x
x
Problems 3–30
Solve
Problems 3–10
exact
3. __ 4. __
5. __ 6.
7. __ 8. __
Problems 11–14
approximate
Problems 15–24
exact
Problems 25–30
approximate
Problems 31–34
approximate
Problems 35–38
y
_____
Problems 39–48
exact
Problems 46–48
Problems 49–54
Problems 49 and 50
1.
_ _
__ _ _
2.
3. r
x
x r?

TEAChER’SEDITION
Section 6-5:
Circular Functions
circular functions,
Learn about the circular functions and their relationship to trigonometric
functions.
y y x
x
FiguFiguFi re 6-5a
x xy
uv u, vvv
x
x
x x
ux
x
xx
xv
v
x
x
x u
x
FiguFiguFi re 6-5b
Circul6-5
The normal hual hual man EKGEKGEK
(electrocadiogram) is periodic.
Learn
funct
Objective
307
Section 6-5
PLANNING
Class Time
Homework Assignment
Teaching Resources
x x x
TEACHING
Section 6-5: Circular Functions

Teacher’sEdition
x x
x x x x
x
x
standard position
u, v
x
x u u
x ________________ v__ v
circular function x
DEFINITION: Circular Functions
u, v x
, circular functions x
x v x u
x x
x
v__
u x x
x
u__
v
x x
__
u x
x
__
v
x
v
u
v
x
x
x, x u, v
u
x
x
Figure 6-5c
Section Notes
circular functions.
,
Note:
Problems 46–48
Exploration Notes
Exploration 6-5a
Degrees Radians
Phrase
Variable x
Symbol

TeacheR’sEdition
309Section 6-5: Circular Functions
y x
x
x x
,
Note: MAXIMUM
x
x, ,
y
x
Figure 6-5e
y C A B x D x
.
y C A B C D
A
B . B
x D
y x
EXAMPLE 1
SOLUTION
y
x
Figure 6-5d
EXAMPLE 2
ySOLUTION
309
Problem 48:
x x x
x x x
Exploration 6-5a:
Technology Notes
Problem 45:
Problem 46:
www.keymath.com/precalc,

TEAChER’SEDITION
Chapter 6:
y x.
x
x ,, ,,
x
x
x
xxx x
EXAMPLE 3
SOLUTION
y
x
FiguFiguFi re 6-5f
ReadingAnalysis
circulararar trigonometric
Quick Review
Q1.
Q2.
Q3.
Q4.
Q5.
Q6.
Q7.Q7.Q7 y
Q8. yyy
Q9.
Q10.
1. 2.
3. 4.
5. 6.
7. 8.
9. 10.
11. 12.
5min
Reading Analysis Q9
Problem Set 6-5
Technology Notes (continued)
Activity:
Instructor’s Resource
Book
Activity:
CAS Suggestions
PROBLEM NOTES
www.keypress.com/
keyonline.
Q1. Q2.
Q3. Q4.
Q5. Q6.
Q7. Q8.
Q9. Q10.
Problems 1–24
1. __ 2. __
3. __ 4. __
5. 6.
7. 8.
Problems 9–12
9. __ 10.
11. 12.
13. 14.
15. 16.
17. 18.
19. 20.
Problems 21–24
21.
__
____ 22. ______
23. ______ 24.

TeacheR’sEdition
13. 14.
15. 16.
17. 18.
19. 20.
21. 22.
23. 24.
25. y x
26. y x
27. y x
28. y x
29. y x 30. y x
31. y x 32. y x
33. y
x
34. y
x
35.
36.
37.
38.
y
x
y
x
y
x
y
x
311
27.
y
x
2
4
28.
y
x
2
1
29. n
n
x
y
4
4
30. __ __ __n
__n
x
y
4
1
Problems 33–42
Problems 41 and 42
33. y __ x
34. y x
35. y ___ x
36. y __ x
37. y __x
38. y __x
Problems 25–32
25.
y
x
4
5
26.
y
x
4
9

TEAChER’SEDITION
Chapter 6:
39.
40.
41.
42.
43.
zzz t
t
44.
EEE r
r
45. Sinusoid TranTranTr slation Problem:
y x
y xxx
y
x
FiguFiguFi re 6-5g
a.
x
b. y x
c.
d. x n xxx
n
e.
Sinusoid TranTranTr slation
y xxx
y x k k
k
46. Sinusoid Dilation Problem:
uv
xxx xxx
uv
xy y x
y xxx
y
x
y
x
z
t
E
r
39. y x
40. y x
41. z t
42. E ___ r
Problems 43 and 44
43. z
z z
44. E
E E
Problem 45a
Problem 45b
x
45a. __
x x __
45b.
45c.
45d.
45e. k
Problem 46

TeacheR’sEdition
v or y
u or x
2 3 4 5
1
1
2x
x
(x2
, y2
)
(x1
, y1
)
(u2, v2)
(u1
, v1
)
x
Figure 6-5h
a. Explain why the value of v for each angle is
equal to the value of y for the corresponding
sinusoid.
b. Create Figure 6-5h with dynamic geometry
software such as Sketchpad, or go to
www.keymath.com/precalc and use the
Sinusoid Dilation exploration. Show the whole
unit circle, and extend the x-axis to x 7. Use
a slider or parameter to vary the value of x. Is
the second angle measure double the first one
as x varies? Do the moving points on the two
sinusoids have the same value of x?
c. Replace the 2 in sin 2x with a variable factor,
k. Use a slider or parameter to vary k. What
happens to the period of the (solid) image
graph as k increases? As k decreases?
47. Circular Function Comprehension Problem:
For circular functions such as cos x, the
independent variable, x, represents the length of
an arc of the unit circle. For other functions you
have studied, such as the quadratic function
y ax2
bx c, the independent variable, x,
stands for a distance along a horizontal number
line, the x-axis.
a. Explain how the concept of wrapping the
x-axis around the unit circle links the two
kinds of functions.
b. Explain how angle measures in radians link
the circular functions to the trigonometric
functions.
48. The Inequality sin x x tan x
Problem: In this problem you will examine the
inequality sin x x tan x for 0 x __
2.
Figure 6-5i shows angle AOB in standard
position, with subtended arc AB of length x on
the unit circle.
O C A
DB
x
Figure 6-5i
a. Based on the definition of radians, explain
why x is also the radian measure of
angle AOB.
b. Based on the definitions of sine and tangent,
explain why BC and AD equal sin x and tan x,
respectively.
c. From Figure 6-5i it appears that
sin x x tan x. Make a table of values
to show numerically that this inequality is true
even for values of x very close to zero.
d. Construct Figure 6-5i with dynamic geometry
software such as Sketchpad, or go to
www.keymath.com/precalc and use the
Inequality sin x x tan x exploration. On
your sketch, display the values of x and the
ratios (sin x)/x and (tan x)/x. What do you
notice about the relative sizes of these values
when angle AOB is in the first quadrant? What
value do the two ratios seem to approach as
angle AOB gets close to zero?
49. Journal Problem: Update your journal
with things you have learned about the
relationship between trigonometric functions
and circular functions.
313
x
47b.
m
m ___
Problem 48
x
BC, x
x
48a. m AOB ______ x__ x
48b.
BC x, AD AD___
OA
x
48c.
48d. x____
x
x x____
x
x
49.
1.
2.
3.
x sin x tan x
46a.
y x
v__________ v__ v
y x
v__________ v__ v
46b.
x
46c. k
__
|k|
Problem 47
47a. x
x
x

TEAChER’SEDITION
Chapter 6:
Inverse Circular Relations:
Given y, Find x
yyy x
xxx y
xxx y
yyy xxx
The Inverse Cosine Relation
arccosine,
n
n
n
n
Invers
Given
6-6
Radar speed guns use inverse
relations to calculate the speed of
a car from time measurements.
Objective
v
u
x
x
u
FiguFiguFi re 6-6a
Section 6-6
PLANNING
Class Time
Homework Assignment
Teaching Resources
yyy
x
yyy x
y, x
TEACHING
Section Notes
arccosine,
1. __
2. ___
3. ___
4. ___
function,
__
relation, infinite
,,

TeacheR’sEdition
315Section 6-6: Inverse Circular Relations: Given y, Find x
DEFINITION: Arccosine, the Inverse Cosine Relation
x x n x x n
n
Verbally:
Note: x principal value
circular
n
n
,
,
, ,
,
, , ,
,
Note:
Finding x When You Know y
y
x y
x
y
x
Figure 6-6b
graphically x y
x
x
EXAMPLE 1
SOLUTION
EXAMPLE 2
SOLUTION
___ x
__
both
n
n
n
f x x
f x x
x(or n) f1
(x) f2
(x)
n.
n
collection
Exploration 6-6a
x Exploration 6-6b
x
y
both
x
x

TEAChER’SEDITION
Chapter 6:
numericallycallycallyy xxx
fff xxx x
fff xxx
x x
x
x
x
x
algebraicallalgebraicallalgebr yaicallyaicallyy xxx
x
x
x
x
x n
x n
x n
x n n
x
EXAMPLE 3
fff xSOLUTION
y
x
x
EXAMPLE 4
SOLUTION
x
n
x
x
Point out the mathematical meanings
of particular,particular,particular principal, and
supplementary. Bilingual dictionaries
may give only the standard English
definitions.
There are several different ways to
write 2 n and 360°n. Students may
have learned to write this as 2k oror
2z and 360°and 360°k or 360°z.
Be aware that some students who
have studied trigonometry (to varying
levels) may have been taught to leave
answers (such as those in Example 4)
in fraction form. Confirm that
students understand your expectation
regarding how to write their answers.
The Reading Analysis may be
confusing for students who have
studied trigonometry from a different
approach.
Exploration Notes
Exploration 6-6a asks questions based
on a given sinusoidal graph. You can
use this exploration instead of doing the
examples with your class. The examples
can reinforce what students learned
through the exploration. Students find
values graphically, numerically, and
algebraically. This exploration can be
used as a small-group introduction to the
section, as a daily quiz after the section
is completed, or as a review sheet to be
assigned later in the chapter. If you use
the exploration to introduce the section,
make sure students have written a correct
equation for Problem 2 before they work
on the other problems. Allow about
20 minutes for the exploration.
Exploration 6-6b has students find the
same x-values for the same function
as in Exploration 6-6a, but using
algebraic methods rather than graphical
ones. Allow 15–20 minutes for this
exploration.
Technology Notes
Exploration 6-6a: Sinusoids, Given y,
Find x Numerically in thex Numerically in thex Instructor’s
Resource Book can be done with the aid
of Sketchpad or Fathom.
CAS Suggestions
Example 1 can be solved efficiently using
the command Solve(cos x = –0.3, x). Note
the return of the symbol n# (where # is anyn# (where # is anyn#
integer) in the CAS result.

TEAChER’SEDITION
Section 6-6:
Notes:
n, n xxx
xxx
n n
xxx n
ReadingAnalysis
Quick Review
Q1.
y x
Q2.
y
Q3.
Q4.
Q5. y
Q6. y
Q7.Q7.Q7
Q8. x yy
Q9.
Q10.
1. 2.
3. 4.
a. x
y
b.
c. x
d. x
e. xxx
y
5. y
y
xx
6.
y
y
xx
5min
Reading Analysis
Problem Set 6-6
PROBLEM NOTES
Supplementary Problems designed
to introduce arcsine and arctangent
keyonline.
Q1. __
2
Q2. 90
Q3. 30 Q4. __
4
Q7. tan 1 3__
7
23.1985...
Q8. Circle of radius 3 and center (0, 0)
Q9. y abx
, a 0, b 0
Q10. Periodic
Problems 1–4 provide students with
practice solving inverse circular relation
equations. Because no method is
specified, students may find the solutions
numerically, graphically, or algebraically.
Problems 1–4 can be solved on a
CAS using the method suggested in the
CAS Suggestions.
Parts c of Problems 5–12 can be
approached as described in the CAS
Suggestions. Using appropriate domain
restrictions, students can calculate all of
the solutions simultaneously.
Problems 5–10 require students to
write equations for sinusoids and use
graphical, numerical, and algebraic
methods to find x-values corresponding
to a particular y-value. These problems
are similar to the examples and to
Exploration 6-6a. In part a, students
should estimate the graphical solutions
from the graph and not use the intersect
feature on their grapher.
One way to answer part e of
Problems 5–10 is to restrict x to valuesx to valuesx
between 100 and 100 one period.
Based on the cycle of the graph, there
should be two answers, both of which are
calculated when using a CAS.
To list the values simultaneously, convert
the previous answer line into list format
and use a condition at the end of the line for
the random integer n9 to assume the valuesn9 to assume the valuesn9
0, 1, 2, and 3—the first four non-negative
integers. The command should look like
this:
{2*n9*{2*n9*{2*n9* + 1.8754889808103, 2*n9*+ 1.8754889808103, 2*n9* –
1.8754889808103} n9 = {0, 1, 2, 3}
A similar approach can be used in
Examples 3 and 4.
To find a specific intersection among
the infinite solutions to an inverse
trigonometric relation, it is helpful to
state constraints after the Solve command.
The graph in Problem 5 seems to have,
among others, an intersection with y 6
somewhere between x 20 and x 23. To
find this intersection, students can enter
Solve(2 + 5 cos(Solve(2 + 5 cos(Solve(2 + 5 cos( /10(x –/10(x – 3)) = 6, x) 20 < x < 23
into their CAS.
See pages 1005–1006 for answers to
Problems Q5, Q6, and 1–6.

Teacher’sEdition
7. y 1
y
y 1
x
2
6
0.3 4.3
8. y 2
y
x
4
2
6.70.7
y 2
9. y 1.5
y
x
17
2
4
y 1.5
10. y 4
y
x
13 2
7
2
3
y 4
For the trigonometric sinusoids graphed in
Problems 11 and 12,
a. Estimate graphically the first three positive
values of for the indicated y-value.
b. Find a particular equation for the sinusoid.
c. Find the -values in part a numerically, using
the equation from part b.
d. Find the -values in part a algebraically.
11. y 3
y
10
2
y 3
150° 330°
12. y 5
y
6
2
y 5
10° 100°
13. Figure 6-6c shows the graph of the parent cosine
function, y cos x.
a. Find algebraically the six values of x shown on
the graph for which cos x 0.9.
b. Find algebraically the first value of x greater
than 200 for which cos x 0.9.
y
1 y 0.9
x
1
2 3 4 5
Figure 6-6c
7a. x
7b. y __ x
7c. x
7d. x
7e. x
8a. x
8b. y __ x
8c. x
8d. x
8e. x
9a. x
9b. y __ x
9c. x
9d. x
9e. x
10a. x
10b. y ___ x
10c. x
10d. x
10e. x
Problems 11 and 12
Problems 11 and 12, Problems 5–10
11a.
11b. y
11c.
11d.
12a.
12b. y __
12c.
12d.
Problem 13
x
x
13a. x
13b. x

TEAChER’SEDITION
Section 6-7:
Sinusoidal Functions as
Mathematical Models
Given a verbal description of a periodic phenomenon, write an equation
using the sine or cosine function and use the equation as a mathematical
model to make predictions and interpretations about the real world.
Sinuso
Mathe
6-7
FiguFiguFi re 6-7a
Given
using
Objective
1.
t
c
t
2.
3.
4.
1 3
EXPLORATION 6-7: Chemotherapy Problem
319Section 6-7: Sinusoidal Functions as Mathematical Models
Section 6-7
PLANNING
Class Time
Homework Assignment
Day 1:
Day 2:
Teaching Resources
TEACHING
Section Notes
Exploration 6-7
Exploration Notes
Exploration 6-7 Instructor’s Resource Book
2. c __ __
4.

Teacher’sEdition
Waterwheel Problem:
P d
a. d t,
b. d t,
c. P
t P
d. t P
a.
d
d
d
t
t
b. d C A B t D
C A
D
B
d t
c.
d
P
WaEXAMPLE 1
d
P
Figure 6-7b
a.SOLUTION
d
t
Figure 6-7c
d
t
B
Figure 6-7d
t
d
t
d
t
d
t
d
enters
not emerges
x y
y x
almost all
x
waterwheel, emerge,
chemotherapy, red blood cell count, Problem 1
Exploration 6-7a
x
Technology Notes
Example 1:

TEAChER’SEDITION
Section 6-7:
d. PPP
d t
t
www.keymath.com/precalc, Waterwheelheelheel
ddd t.
ReadingAnalysis
Quick Review
y x
Q1.
Q2.
Q3.
Q4. y
Q5.
Q6. y
Q7.Q7.Q7 x y
Q8. x
Q9. x
Q10. yy x
x
yyy
1. Steamboat Problem:
d,
t,
a.
b.
c.
d.
e. tt
f.
5min
Reading Analysis 1 Steamboat Problem:
Problem Set 6-7
CAS Suggestions
By this point in the chapter, students
should have explored and begun to
master the techniques and implications
of circular functions. Because the
objective of this section is to apply the
mathematics, not perform the algebraic
manipulations, students benefit from
using a CAS, which allows them to focus
on what the mathematics mean.
Defining the equation in part b of
Example 1 and using that definition to
graph and evaluate the model can help
students solve part c. TI-Nspire users
may want to use a word or abbreviation
to define the function. These figures
show the definition of the waterwheel’s
height. The t-intercepts for part d can be
found using either the Solve or the Zeros
command.
PROBLEM NOTES
Supplementary Problems for this section
keyonline.
Q1. 5 Q2. 12
Q3. 1___
12
Q4. 4
Q5. 7 Q6. 9
Q7. 6.5 Q8. 1, 13, 25
Q9. __
3
Q10. 9
Exploration 6-7: Chemotherapy
Problem can be done with sliders in
Sketchpad or Fathom.
Exploration 6-7a: Oil Well Problem
in the Instructor’s Resource Book can
be done with sliders in Sketchpad or
Fathom.
CAS Activity 6-7a: Epicenter of
an Earthquake in the Instructor’s
Resource Book has students explore
what information is required in order
to find the epicenter of an earthquake.
Students find the epicenter of an
earthquake using triangulation. Allow
25–30 minutes.
See page 1006 for answers to Problem 1.

TEAChER’SEDITION
Chapter 6:
2. Fox PopulaPopulaPo tion Problem:
t
t
t
a.
b.
c. t
d.
ttt
e.
3. Bouncing Spring Pring Pring oblem:
a.
b.
c.
d.
e.
FiguFiguFi re 6-7e
4. Rope Swing Pring Pring oblem:
x
y
x
y
yyy x
yyyy
FiguFiguFi re 6-7f
a. yyy xxx
Solve
Solve
Zeros
Solve
Problem 2
2a., 2e.
2b. F ___ t
2c. F F
F F
2d. t
Problem 3
3a.
3b. d ___ t
3c. d
3d. d
3e. t
t
800
4
F
200
15105
20
40
60
t
d
Problem 4 y 4a.
y __ x
5
y
x
20
20

TeacheR’sEdition
b. y x
c.
y
d.
y
5. Roller Coaster Problem:
a.
y
x
y x
b.
c.
d.
y
x
Figure 6-7g
6. Buried Treasure Problem:
x
y
x
y
x
Figure 6-7h
a. y
surface
x
b
c.
5b.
5c.
x Length
0 m 27 m
2 m 26.8817... m
4 m 26.5287... m
6 m 25.9466... m
8 m 25.1446... m
10 m 24.1352... m
12 m 22.9345... m
14 m 21.5613... m
16 m 20.0374... m
18 m 18.3866... m
20 m 16.6352... m
22 m 14.8107... m
24 m 12.9418... m
26 m 11.0581... m
28 m 9.1892... m
30 m 7.3647... m
32 m 5.6133... m
34 m 3.9625... m
36 m 2.4386... m
38 m 1.0654... m
y Length
0 m 39.7583... m
2 m 36.6139... m
4 m 33.9530... m
6 m 31.5494... m
8 m 29.2961... m
10 m 27.1284... m
12 m 25 m
14 m 22.8715... m
16 m 20.7038... m
18 m 18.4505... m
20 m 16.0469... m
22 m 13.3860... m
24 m 10.2416... m
26 m 5.8442... m
4b. y 16.3826... ft; Zoey was over land.
4c. x 0.3562... s
4d. y 3, the sinusoidal axis
Problem 5 asks students to calculate the
lengths of the horizontal and vertical
support beams needed for a roller coaster.
Some students double the horizontal beam
lengths because the diagram shows only
half of a cycle.
5a. y 12 15cos ___
50
x
5b., 5c. See tables at right.
5d. Vertical timbers 324 m;
horizontal timbers 331 m
Problem 6 shows only half of a cycle.
6a. y 50 100cos ___
600
(x 400)
6b. 50 100cos ___
600
(0 400) 0 m
6c. The vertical tunnel is shorter.

Teacher’sEdition
7. Sunspot Problem: For several hundred
years, astronomers have kept track of the
number of sunspots that occur on the surface of
the Sun. The number of sunspots in a given year
varies periodically, from a minimum of about
10 per year to a maximum of about 110 per year.
Between 1750 and 1948, there were exactly
18 complete cycles.
a. What is the period of a sunspot cycle?
b. Assume that the number of sunspots per year
is a sinusoidal function of time and that a
maximum occurred in 1948. Find a particular
equation expressing the number of sunspots
per year as a function of the year.
c. How many sunspots will there be in the
year 2020? This year?
d. What is the first year after 2020 in which
there will be about 35 sunspots? What is the
first year after 2020 in which there will be a
maximum number of sunspots?
e. Find out how closely the sunspot cycle
resembles a sinusoid by looking on the
Internet or in another reference.
8. Tide Problem: Suppose you are on the beach at
Port Aransas, Texas, on August 2. At 2:00 p.m.,
at high tide, you find that the depth of the water
at the end of a pier is 1.5 m. At 7:30 p.m., at low
tide, the depth of the water is 1.1 m. Assume that
the depth varies sinusoidally with time.
a. Find a particular equation expressing depth as
a function of the time that has elapsed since
12:00 a.m. August 2.
b. Use your mathematical model to predict the
depth of the water at 5:00 p.m. on August 3.
c. At what time does the first low tide occur on
August 3?
d. What is the earliest time on August 3 that the
water depth will be 1.27 m?
e. A high tide occurs because the Moon is
pulling the water away from Earth slightly,
making the water a bit deeper at a given point.
How do you explain the fact that there are two
high tides each day at most places on Earth?
Provide the source of your information.
9. Shock Felt Round the World Problem: Suppose
that one day all 300 million people in the
United States climb up on tables. At time
t 0, they all jump off. The resulting shock wave
starts Earth vibrating at its fundamental period,
54 min. The surface first moves down from its
normal position and then moves up an equal
distance above its normal position (Figure 6-7i).
Assume that the amplitude is 50 m.
Up 50 mDown 50 m
50 m 50 m
Jump!
Figure 6-7i
Problem 7 asks students to consult a
reference in order to check the accuracy
of the model. One possible reference is
the September 1975 issue of Scientific
American.
7a. 11 yr
7b. S 60 50 cos 2___
11
(t 1948)
7c. S(2020) 12 sunspots
7d. S(2021) 27 sunspots; S(2022)
53 sunspots; maximum in 2025.
7e. The sunspot cycle resembles a
sinusoid slightly but is not one.
Problem 8 requires students to convert
date and time information to the number
of hours since midnight on August 2.
8a. d 1.3 0.2cos ___
5.5
(t 14)
8b. 1.1 m
8c. 6:30 a.m.
8d. 4:00:49 a.m.
8e. On the side closest to the Moon, the
water is pulled more than Earth, causing
a high tide. On the opposite side, farthest
from the Moon, Earth is pulled more
than the water, causing another high
tide.
Problem 9 is fairly straightforward and
usually does not present trouble for
students. However, some students do
not realize that at time 0 the vertical
displacement of Earth is also zero; they
may start the graph at a minimum value
instead of at zero.

TeacheR’sEdition
a.
b.
c.
d. t
e.
10. Island Problem:
x
y x
x y
y
x
x
Figure 6-7j
a.
y
b.
y
c.
y
x
d.
x
e.
f. x
11. Pebble in the Tire Problem:
a.
b.
c.
d.
9a.
9b. t
9c. d ___ t
9d. d
9e. t
Problem 10
10a. y x
10b. y x
10c.
x
10d.
10e.
y
10f. x
Problem 11
11a.
11b. y x___
11c.
11d. x
50
50
50
t (min)
d (m)
100
(91.81, 40)
(808.19, 40)
x
y
50 100
12
24
y
x

Teacher’sEdition
12. Oil Well Problem: Figure 6-7k shows a vertical
cross section through a piece of land. The
y-axis is drawn coming out of the ground at the
fence bordering land owned by your boss, Earl
Wells. Earl owns the land to the left of the fence
and is interested in acquiring land on the other
side to drill a new oil well. Geologists have found
an oil-bearing formation below Earl’s land that
they believe to be sinusoidal in shape. At distance
x 100 ft, the top surface of the formation
is at its deepest, y 2500 ft. A quarter-cycle
closer to the fence, at distance x 65 ft, the
top surface is only 2000 ft deep. The first 700 ft of
land beyond the fence is inaccessible. Earl
wants to drill at the first convenient site beyond
x 700 ft.
a. Find a particular equation expressing y as a
function of x.
b. Plot the graph on your grapher. Use a window
with 100 x 900. Describe how the
graph confirms that your equation is correct.
c. Find graphically the first interval of x-values
in the available land for which the top surface
of the formation is no more than 1600 ft deep.
d. Find algebraically the values of x at the ends of
the interval in part c. Show your work.
e. Suppose the original measurements were
slightly inaccurate and the value of y shown
at 65 ft is actually at x 64 ft. Would this
fact make much difference in the answer to
part c? Use the most time-efficient method to
arrive at your answer. Explain what you did.
13. Sound Wave Problem: The hum you hear on
some radios when they are not tuned to a station
is a sound wave of 60 cycles per second.
Bats navigate and communicate using ultrasonic
sounds with frequencies of 20–100 kilohertz (kHz),
which are undetectable by the human ear. A kilohertz
is 1000 cycles per second.
a. Is 60 cycles per second the period, or is it
the frequency? If it is the period, find the
frequency. If it is the frequency, find the
period.
b. The wavelength of a sound wave is defined
as the distance the wave travels in a time
interval equal to one period. If sound travels
at 1100 ft/s, find the wavelength of the
60-cycle-per-second hum.
c. The lowest musical note the human ear
can hear is about 16 cycles per second. In
order to play such a note, a pipe on an organ
must be exactly half as long as the wavelength.
What length organ pipe would be needed to
generate a 16-cycle-per-second note?
y
x
Fence
Oil-bearing
formation
Top surface
y 2000 ft
x 700 ft
100 65 Available land
y 2500 ft
Inaccessible land
Figure 6-7k
Problem 12 is not difficult if students
realize that half of a cycle occurs between
the x-values of 30 and 100.
12a. y 2000 500cos ___
70
(x 30)
12b.
The graph matches the description and
figure.
12c. Roughly 795 ft x 825 ft
12d. Roughly 796 ft x 824 ft
12e. The new interval is roughly
700 ft x 707 ft. This is a very small
region to drill.
Problem 13 provides practice with
frequency and is quite easy. It does not
require students to write a sinusoidal
equation. You may wish to add this
part d to the question: “Research in the
library or on the Internet the physics of
organ pipes. You might try looking up
“The Physics of Organ Pipes,” by Neville
Fletcher and Suzanne Thwaites, in the
January 1983 issue of Scientific American.
Describe the results of your research in
your journal.”
13a. Frequency 60 cycles/s;
period 1___
60
s
13b. Wavelength 220 in.
13c. 34 ft 4.5 in.
y
x
100 100
2000
200 300 400 500 600 700 800 900
1000

TeacheR’sEdition
14. Sunrise Project:
t d
d
a.
d
b.
c.
d.
e.
varies
t
d
Figure 6-7l
15. Variable Amplitude Pendulum Project:
y A Bt
A,
A a bt
y
Figure 6-7m
a, b, B
t
Problem 14
14a. t ___ ___ ___ d
14b.
14c.
14d.
14e.
t __
__ ___ d ___ d
Problem 15
15.

TEAChER’SEDITION
Chapter 6:
Rotary Motion
When you ride a merry-go-round, you go faster when you sit nearer the outside.
As the merry-go-round rotates through a certain angle, you travel farther in the
same amount of time when you sit closer to the outside (Figure 6-8a).
P
P1
2
Rotation
Farther
(so faster)
Shorter
(so slower)
FiguFiguFi re 6-8a
However, all points on the merry-go-round turn through the same number of
degrees per unit of time. So there are two differenfferenff t kinds of speeof speeof d, or velocity,ty,ty
associated with a point on a rotating object. The angular velocity isty isty the number of
degrees or radians per unit of time, and the linear velocity is the distance per unit
of time.
In this exploration you will practice computing and interpreting linear and
angular velocities.
Rotary
When you rid
6-8
Objective
Rotate
The figure shows a rotating ruler attached by
suction cup to a dry-erase board. Marking pens are
put into holes in the ruler, and the ruler and pens
are rotated slowly from the initial position (dotted),
reaching the final position (solid) after 5 s. Each
pen leaves an arc on the board.
1. With a flexible ruler you find that the curved
lengths of the arcs are 60 cm (outer) and
24 cm (inner). At what average speed (cm/s)
did each pen travel in the 5 s? Which pen
moved faster?
2. Suppose the angle from the initial position to
the final position is 115°. At what number of
degrees per second did each pen move? Did
one pen move more degrees per second than
the other?
3. Suppose the inner pen is 12 cm from the
center of rotation and the outer pen is 30 cm
from the center of rotation. An angle of 115°
is about 2 radians. Show that the distance
each pen moved can be found by multiplying
the radius of the arc by the angle measure in
radians.
R did h t l i th 5 ? Whi h
EXPLORATION 6-8: Angular and Linear Velocity
continued
Section 6-8
PLANNING
Class Time
Homework Assignment
Day 1:
Day 2:
Teaching Resources
TEACHING
Exploration Notes
Exploration 6-8
1.
2.
3.
4.
5.
6.
7.

TeacheR’sEdition
329Section 6-8: Rotary Motion
EXPLORATION, continued
4.
linear velocities.
angular velocity.
5.
6.
7.
r
a
v
t
r
r
a t
v a/t
/t
Figure 6-8b
DEFINITIONS: Angular Velocity and Linear Velocity
angular velocity, ,
linear velocity, v,
t v a
t
Section Notes
revolutions per minute
rpm
rev/min
Angle
(radians)
Arc Length
(cm)
Time
(s)
Linear Velocity
(cm/s)
Angular Velocity
(rad/s)
Outer ___ __
Inner ___ __

Teacher’sEdition
a r
a
t
r
t r t
v,
r .
v r
PROPERTIES: Linear Velocity and Angular Velocity
a r
v r
Analysis of a Single Rotating Object
a.
b.
c.
a.
b.
v r
EXAMPLE 1
Figure 6-8c
a.SOLUTION
revolutions
per minute
w
LP.
axle sprocket
Exploration 6-8a
Exploration 6-8b

TEAChER’SEDITION
Section 6-8:
c.
v rrr
Analysis of Connected Rotating Objects
a.
b.
c.
d.
e.
a.
b. v rrr
FiguFiguFi re 6-8d
EXAMPLE 2
a.SOLUTION
CAS Suggestions
Solve

TEAChER’SEDITION
Chapter 6:
c. v
d. v rrr v
r
v rrr
e. v rrr
v rrr
CONCLUSIONS: Connected Rotating Objects
.
ReadingAnalysis
Quick Review
Q1.
Q2.
Q3. xxx
Q4. y xxx
yyy
Q5.
y x
Q6. fff
g xxx f xxx
Q7.Q7.Q7
Q8.
Q9. x x
Q10.
5min
Reading Analysis Q4. y xx
Problem Set 6-8
PROBLEM NOTES
Q1.
Q2.
Q3.
Q4.
Q5.
Q6.
Q7.
__
Q8.
Q9. x x
Q10.

TEAChER’SEDITION
Section 6-8:
1.
a.
b.
2.
a.
b.
3.
a.
b.
c.
4.
a.
b.
c.
5.
a.
b.
c.
6.
a.
b.
7.
a.
b.
c.
Problems 1–17
Problems 1–8
1a.
1b. ___
2a.
2b. v
v
Problem 3c
3a.
3b.
3c.
v
4a.
4b.
4c.
5a. ___
5b.
5c. ___
v
6a. v
6b. v
7a. __
7b. __
7c. v _____

Teacher’sEdition
d.
zero
e.
8. Paper Towel Problem:
a.
b.
c.
d.
e.
9. Pulley Problem:
Figure 6-8g
a.
b.
c.
d.
e.
f.
10. Gear Problem:
Figure 6-8h
a.
b.
c.
d.
e.
f.
11. Tractor Problem:
Figure 6-8i
a.
b.
7d.
v __
7e. v
8a. v
8b. v
8c.
8d.
8e.
Problems 9–17
9a. ____
9b. v ____
9c. v ____
9d. v ____
9e. ____
9f. ____
v
Problem 10
10a.
10b. v
10c. v
10d.
10e.
10f.
d_____
d
11a. v ___
v ____
11b. v ___

TeacheR’sEdition
c.
12. Wheel and Grindstone Problem:
Figure 6-8j
a.
b.
c.
13. Three Gear Problem:
Figure 6-8k
a.
b.
c.
d.
e.
f.
14. Truck Problem:
Figure 6-8l
a.
b.
c.
d.
mile per hour
15. Marching Band Formation Problem:
a.
11c.
12a. v
12b.
12c.
13a.
13b.
13c. v
13d. v
13e. ____
13f. v ____
14a.
14b. v
14c.
14d. v
15a. ___ _____

Teacher’sEdition
b.
c.
d.
e.
Figure 6-8m
16. Four Pulley Problem:
Figure 6-8n
a.
b.
c.
d.
e.
f.
17. Gear Train Problem:
Figure 6-8o
a.
b.
c.
d.
e. reduction ratio
15b. __
15c.
15d.
15e.
____________________
16a.
16b. v
16c. v
v
v
16d. v
16e.
16f. _____ _____
17a.
17b. v
v
v
17c. v
17d.
17e. _____ _____

TEAChER’SEDITION
Section 6-9:
Chapter Review and TestChapte6-9
R0.
R1. a.
b. y
R2. a.
y
b.
y
FiguFiguFi re 6-9a
c.
y
FiguFiguFi re 6-9b
d.
e.
R0. b.
Review Problems
337Section 6-9: Chapter Review and Test
Section 6-9
PLANNING
Class Time
Homework Assignment
Day 1:
Day 2
Teaching Resources
TEACHING
Section Notes
R0.
R2a.
R2b. y ___
y ___
R2c. y ___
R2d.
R2e. ___ ___
y
90°
3

Teacher’sEdition
R3. a. y
b.
c. y
d.
e.
f. y
R4. a.
b.
c.
d.
e.
R5. a. uv
x
u v
x
x
b.
c.
d.
e.
f.
y x y x
g.
y x
h.
y
x
Figure 6-9c
R6. a.
b.
c.
d.
x y
e. x
y
y
x
y
Figure 6-9d
Problem C3
pendulum, tide,
_
Exploration Notes
Exploration 6-9a
Exploration 6-9b
Technology Notes
Exploration 6-9a Instructor’s
Resource Book

TEAChER’SEDITION
Section 6-9:
R7.
a.
b.
c.
d.
R8.
a.
b.
c.
d.
e.
f.
g.
h.
i.
j.
C1. a.
b.
C1. a.
Concept Problems
PROBLEM NOTES
Problem R2b
R6a. n
R6b.
R6c.
R6d. x
y __ x
x
x
R6e. x
R7b. y __ t
R7c. t y
R7d. t
R8a. ____
R8b. ___
R8c. ___
___
____ ____
R8d. v
R8e. v
R8f. v
R8g.
R8h.
R8i. v
R8j. v
Problem C1
R4a. __ __
__
R4b.
R4c.
R4d. __
R4e.
R5a.
R5b. __
1
2
3
1
2
3 u
v
r 1

Teacher’sEdition
c. d
d
P
Figure 6-9f
d. t
P
P
P
e. d
t
f. P
t
g. P
h.
C2. Inverse Circular Relation Graphs:
a.
y x
x x
x
x y
b.
function
relation, y x,
x y
t
:
x t
y t
t x
c.
d.
y x
e.
y x
x
y
Figure 6-9g
f.
g.
not
h.
y x
C1c.
C1d.
C1e. d ___ t
C1f.
C1g.
C1h.
Problem C2
C2a.
C2b.
C2c.
C2d.
__ t __
C2e. x t, y t.
y
x
1
2
5
10
2 4 86
d
t
y
x
1
2
y
x
1
2
C2f.
y x.
C2g.
y x
y x
C2h.
y x
y
x
1
2

TeacheR’sEdition
i.
j.
C3. Merry-Go-Round Problem:
4 ft
Figure 6-9h
a.
combined
b.
c.
t
t,
d.
e.
Part 1: No calculators allowed (T1–T9)
T1. x
uv
x
x
v
x
u
Figure 6-9i
T2.
T3.
T4.
T5.
T6.
f x x
T7.
T8.
Part 1: No calculators allowed (T1–T9) T2.
Chapter Test
C3d.
5 10
5
10
15
20
25
t (s)
d (ft)
C3e.
Problems T1 and T2
Instructor’s
Resource Book.
T1., T2.
T3. ___
T4. ___
T5.
T6.
T7.
T8. v
1
2
3
12
3 u
v
r 1
4
f(x)
x
8
3
C2i.
y x
y
x
1
2
C2j.
C3a. v
v
C3b.
C3c. d t t___ t,
d t

Teacher’sEdition
T9.
Part 2: Graphing calculators allowed (T10–T24)
T10.
x y
y
T11.
y
x
Figure 6-9j
d t
d t
Figure 6-9k
T12.
T13.
T14. t
T15.
x t
x
y d
T16.
T17.
T18.
Bicycle Problem:
Figure 6-9l
T19.
T20.
T21.
T22.
T23.
T24.
T9.
T10.
T11. y __ x
T12. d t n
T13. d t n
T14. t
d
T15.
T16. t
T17. t
T18. t
T19.
T20. v
T21.
T22. v
T23. d t t
1.0
10
20
30
40
50
60
t (s)
d (cm)
T24.
y
x
50
741
d
t
3
10

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