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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 a rock formation at a given point. Such
information can be very
ry useful
useful 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
perfor
rf m calculations for such
rfor
ch wa
wavy rock
formations.
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Precalculus with Trigonometry Course Sampler
69
Chapter 6
Applications of Trigonometric
and Circular Functions
Overview
TEAChER’S EDITION
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Sec tion 6 -1
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72
Th circ
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fu ncti
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st like the
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Sinusoids: Amplitude, Period,
6 -1 Sinuso
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Figure 6-1a shows a dilated and
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graphical features. In this section you
will learn how these features relate to
transfor
sf mations you’ve already learned.
sfor
PRO BLE M N OTES
Problems 2–5, 7, and 9 can be
solved using the approaches described in
the CAS Suggestions.
Phase displacement
(horizontal translation)
Period
2. Amplitude 5
y
Amplitude
plitude
2
Sinusoidal axis
One cycle
180°
2
360°
Fi re 6-1a
Figu
Objective
Learn
earn the meanings of amplitude
amplitude,, period, phase displacement, and cycle
cy of a
sinus
sinusoidal graph.
The absolute value of the vertical dilation
of a sinusoid equals the amplitude.
3. 360 for both functions
4. 120
Exploratory Problem Set 6-1
2. Plot the graph of the transf
transfor
ormed cosine
function y 5 cos . What is the amplitude of
this graph? What is the relationship between the
amplitude and the vertical
rtical dila
dilation of a sinusoid?
y
5. Plot the graph of y cos( 60°). What
transformation is caused by the 60°?
6. The ( 60°) in Problem 5 is called the
ar ment of the cosine. The phase displacement
argu
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?
4. Plot the graph of y cos 3. What is the period
of this transf
transfor
ormed function graph? How is the
3 related
d to the transf
transformation? How could you
calculate the period using the 3?
360°
2
The 3 is the reciprocal of the horizontal
dilation. The period equals 360 divided
by 3.
5.
8. The sinusoidal axis runs along the middle of the
graph of a sinusoid. It is the dashed centerline in
Figure 6-1a. What transformation of the function
y cos x does the location of the sinusoidal axis
indicate?
3. What is the period of the transf
transfor
ormed function
in Problem 2? What is the period of the parent
function y cos ?
2
9. What are the amplitude, period, phase
displacement, and sinusoidal axis location of the
graph of y 6 5 cos 3( 60°)? Check by
plotting on your grapher.
y
2
360°
2
Horizontal translation by 60
6. Phase displacement 60. The phase
displacement equals the horizontal
translation.
7.
y
2
360°
2
Section 6-1:
Vertical translation by 6
1. Amplitude 1
8. The vertical translation corresponds
to the sinusoidal axis.
y
2
2
TEAChER’S EDITION
1. Sketch one cycle of the graph
graph of the parent
parent
sinusoid y cos , starting at 0°. What is
the amplitude of this graph?
360°
9. Amplitude 5; Period 120; Phase
displacement 60; Sinusoidal axis 6
y
8
360°
Section 6-1: Sinusoids: Amplitude, Period, and Cycles
283
Precalculus with Trigonometry Course Sampler
73
General Sinusoidal Graphs
6 -2 Genera
Sec tion 6 -2
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Chapter 6: Applications of Trigonometric and Circular Functions
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Section 6-2: General Sinusoidal Graphs
285
Precalculus with Trigonometry Course Sampler 75
T eac h e R ’ s E dition
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Chapter 6: Applications of Trigonometric and Circular Functions
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Precalculus with Trigonometry Course Sampler
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Section Notes (continued)
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because of the complex language.
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to develop their English skills.
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A
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 V 35°. Allow 20–30 minutes for
the exploration.
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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.
t Have students use their paper graphs
while you go over the examples.
Section 6-2: General Sinusoidal Graphs
289
t Have students write the general
sinusoidal equation definition in their
journals both formally and in their
own words.
t Make sure the distinction on page 290
between a particular equation and the
particular equation is clear.
t Consider using the Reading Analysis
as an in-class discussion. Alternatively,
allow ELL students to do it in pairs.
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.
Section 6-2: General Sinusoidal Graphs
289
Precalculus with Trigonometry Course Sampler 79
T eac h e R ’ s E dition
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Chapter 6: Applications of Trigonometric and Circular Functions
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12.
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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 1.11 sin 10(V 16)
10. y 30 20 cos 360(V 0.3)
11. y 1.7 cos(V 30)
12. y 5000 cos 45(V 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 V.
13. r 7 cos 3B
14. y 0.03 sin 0.9C
15. y 35 15 sin 90V
16. y 7 3 sin 4.5V
y
9
17. y 4 9 sin __
13 (V 60)
18. y 60 100 sin 45V
Problems 19 and 20 require students to
find y-values corresponding to given
angle values.
6a. y 8 10 cos 9(V 4)
6b. Amplitude 10; Period 40;
1
Frequency __
40 cycle/deg;
Phase displacement 4;
Sinusoidal axis 8
6c. y 2.1221... at V 10;
y 6.4356... at V 453
Section 6-2: General Sinusoidal Graphs
293
7b. Amplitude 5; Period 120;
1
Frequency ___
120 cycle/deg;
Phase displacement 10;
Sinusoidal axis 3
7c. y 8 at V 70;
y 1.9931... at V 491
19. y 12.4151... at V 300;
y 2.0579..., 1.9420... below the
sinusoidal axis at V5678
20. y 32.3879... at V 2.5; y 60,
on the sinusoidal axis, at V 328
Problems 21 and 22 require students to
write equations for sinusoids based on
verbal descriptions.
7a. y 3 5 cos 3(V 10)
See page 1002 for answers to
Problems 21 and 22.
Section 6-2: General Sinusoidal Graphs
293
Precalculus with Trigonometry Course Sampler 83
T eac h e R ’ s E dition
11.
Problem Notes (continued)
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295
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Section 6-3: Graphs of Tangent, Cotangent, Secant, and Cosecant Functions
297
Precalculus with Trigonometry Course Sampler 87
T eac h e R ’ s E dition
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Problem 16 asks students to
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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
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299
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1
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Section 6-4: Radian Measure of Angles
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 Q radians is equivalent
to 180 degrees, the conversion factor
Q radians
is ________
, a form of 1. Students should
180 degrees
realize that multiplying a number of
degrees by this factor causes the degrees
to cancel out, leaving a number of
radians.
30
degrees __________
Q radians __1Q radians
_________
1
180 degrees 6
Similarly, multiplying a number of
180 degrees
radians by ________
, another form of 1,
Q radians
lets the radians cancel, leaving a number
of degrees. If the units don’t cancel, the
student has the conversion factor upside
down.
5. Answers will vary.
Section 6-4: Radian Measure of Angles
301
Precalculus with Trigonometry Course Sampler 91
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DEFINITION: Radian Measure of an Angle
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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 .
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Because the circumference of a circle is 2r 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:
m°() 360°
m R() ____
180°
_____
_____
2 ____
or
____ ____
2
m°() 360° 180°
m R()
Solving for m R() and m°(), respectively, gives
180°
m°()
R
and
m°() ____
m R() ____
m ()
180°
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
180
To find the degree measure of , multiply the radian measure by ___
.
EXAMPLE 1 SOLUTION
Convert 135° to radians.
Co
I order to keep the units straight, write
In
each qua
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.
135° x
135 x
180° 180 = 3
135 degrees radians
m R() __________ __________ __ 2.3561... radians
180 degrees 4
1
Notes:
3 If the exact value is called for, leave the answer as _34 . If not, you have the choice
of writing the answer as a multiple of or converting to a decimal.
3 The procedure for canceling units used in Example 1 is called dimensional
analysis. You will use this procedure throughout your study of mathematics.
302
302
Chapter 6: Applications of Trigonometric and Circular Functions
Chapter 6: Applications of Trigonometric and Circular Functions
92 Precalculus with Trigonometry Course Sampler SOLUTION
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Section 6-4: Radian Measure of Angles
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Precalculus with Trigonometry Course Sampler 93
T eac h e R ’ s E dition
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Precalculus with Trigonometry Course Sampler 101
T eac h e R ’ s E dition
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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.
a. Explain why the value of v for each angle is
equal to the value of y for the corresponding
sinusoid.
D
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x
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?
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Figure 6-5i
b. Based on the definitions of sine and tangent,
explain why BC and AD equal sin x and tan x,
respectively.
48c.
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.
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 ax 2 bx c, the independent variable, x,
stands for a distance along a horizontal number
line, the x-axis.
Section 6-5: Circular Functions
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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?
a. Explain how the concept of wrapping the
x-axis around the unit circle links the two
kinds of functions.
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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?
b. Explain how angle measures in radians link
the circular functions to the trigonometric
functions.
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Section 6-5: Circular Functions
313
Precalculus with Trigonometry Course Sampler 103
T eac h e R ’ s E dition
(x 2 , y2 )1
Inverse Circular Relations:
6 - 6 Invers
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Section 6-6: Inverse Circular Relations: Given y, Find x
315
Precalculus with Trigonometry Course Sampler 105
T eac h e R ’ s E dition
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Differentiating Instruction
EXAMPLE 3 TEAChER’S EDITION
t Point out the mathematical meanings
of particular,
particular principal, and
supplementary. Bilingual dictionaries
may give only the standard English
definitions.
t There are several different ways to
write 2Qn and 360°n. Students may
have learned to write this as 2kQ
Q or
2zQ
Q and 360°k or 360°z.
t 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.
t The Reading Analysis may be
confusing for students who have
studied trigonometry from a different
approach.
SOLUTION
"+*
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.
316
106
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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.
numerically
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Chapter 6:
Technology Notes
Exploration 6-6a: Sinusoids, Given y,
Find x Numerically in the 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 any
integer) in the CAS result.
Chapter 6: Applications of Trigonometric and Circular Functions
Precalculus with Trigonometry Course Sampler
Notes:
PRO BLE M N OTES
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Supplementary Problems designed
to introduce arcsine and arctangent
are available at www.keypress.com/
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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.
07'
a. .
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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.
y
+
x
6.
+
y
y
x
Section 6-6:
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 values
0, 1, 2, and 3—the first four non-negative
integers. The command should look like
this:
{2*n9*Q + 1.8754889808103, 2*n9*Q –
1.8754889808103} \ n9 = {0, 1, 2, 3}
A similar approach can be used in
Examples 3 and 4.
TEAChER’S EDITION
Reading Analysis
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
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 values
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.
Solve(2 + 5 cos(Q/10(x – 3)) = 6, x) \ 20 < x < 23
into their CAS.
See pages 1005–1006 for answers to
Problems Q5, Q6, and 1–6.
Section 6-6: Inverse Circular Relations: Given y, Find x
317
Precalculus with Trigonometry Course Sampler
107
Problem Notes (continued)
11. y 3
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150°
8. y 2
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6
x
4
a. Find algebraically the six values of x shown on
the graph for which cos x 0.9.
x
1
7
T eac h er ’ s E dition
b. Find algebraically the first value of x greater
than 200 for which cos x 0.9.
2
y
10. y 4
3
13
y 4
9e. x
100°
13. Figure 6-6c shows the graph of the parent cosine
function, y cos x.
y
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3
4
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10b. yDPT___
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7
For the trigonometric sinusoids graphed in
Problems 11 and 12,
a. Estimate graphically the first three positive
values of for the indicated y-value.
10d. x
b. Find a particular equation for the sinusoid.
c. Find the -values in part a numerically, using
the equation from part b.
10e. x
d. Find the -values in part a algebraically.
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318
Chapter 6: Applications of Trigonometric and Circular Functions
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Chapter 6: Applications of Trigonometric and Circular Functions
108 Precalculus with Trigonometry Course Sampler x
Sinusoidal Functions as
6 -7 Sinuso
Sec tion 6 -7
Mathematical Models
Mathe
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Given a verbal description of a periodic phenomenon, write an equation
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model to make predictions and interpretations about the real world.
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Section 6-7: Sinusoidal Functions as Mathematical Models
319
Precalculus with Trigonometry Course Sampler
109
Section Notes (continued)
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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.
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manipulations, students benefit from
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Exploration 6-7: Chemotherapy
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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
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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.
Supplementary Problems for this section
are available at www.keypress.com/
keyonline.
Q1. 5
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See page 1006 for answers to Problem 1.
Section 6-7: Sinusoidal Functions as Mathematical Models
321
Precalculus with Trigonometry Course Sampler
111
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Section 6-7: Sinusoidal Functions as Mathematical Models
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.
Qx
5a. y 12 15 cos ___
50
5b., 5c. See tables at right.
323
18 m 18.4505... m
20 m 16.0469... m
22 m 13.3860... m
5d. Vertical timbers 324 m;
horizontal timbers 331 m
24 m 10.2416... m
26 m
5.8442... m
Problem 6 shows only half of a cycle.
Q (x 400)
6a. y 50 100 cos ___
600
Q (0 400) 0 m
6b. 50 100 cos ___
600
6c. The vertical tunnel is shorter.
Section 6-7: Sinusoidal Functions as Mathematical Models
323
Precalculus with Trigonometry Course Sampler 113
Problem Notes (continued)
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
2Q(t
7b. S 60 50 cos ___
1948)
11
7c. S(2020) 12 sunspots
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.
7e. The sunspot cycle resembles a
sinusoid slightly but is not one.
T eac h er ’ s E dition
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.
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.
7d. S (2021) 27 sunspots; S (2022)
53 sunspots; maximum in 2025.
Problem 8 requires students to convert
date and time information to the number
of hours since midnight on August 2.
Q(t 14)
8a. d 1.3 0.2cos ___
5.5
8b. 1.1 m
a. Find a particular equation expressing depth as
a function of the time that has elapsed since
12:00 a.m. August 2.
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.
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.
50 m
Jump!
Down 50 m
Figure 6-7i
324
324
Chapter 6: Applications of Trigonometric and Circular Functions
Chapter 6: Applications of Trigonometric and Circular Functions
114 Precalculus with Trigonometry Course Sampler 50 m
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Section 6-7: Sinusoidal Functions as Mathematical Models
325
12
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Section 6-7: Sinusoidal Functions as Mathematical Models
325
Precalculus with Trigonometry Course Sampler 115
T eac h e R ’ s E dition
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Problem Notes (continued)
Problem 12 is not difficult if students
realize that half of a cycle occurs between
the x-values of 30 and 100.
Q(x 30)
12a. y 2000 500cos ___
70
y
12b.
x
100
100 200 300 400 500 600 700 800 900
1000
2000
The graph matches the description and
figure.
12c. Roughly 795 ft x 825 ft
12d. Roughly 796 ft x 824 ft
T eac h er ’ s E dition
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.”
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 100x 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.
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?
Inaccessible land
100 65
x 700 ft
y 2000 ft
y 2500 ft
Top surface
Oil-bearing
formation
Figure 6-7k
13c. 34 ft 4.5 in.
326
Chapter 6: Applications of Trigonometric and Circular Functions
Chapter 6: Applications of Trigonometric and Circular Functions
116 Precalculus with Trigonometry Course Sampler 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.
y Fence
13a. Frequency 60 cycles/s;
1s
period ___
60
13b. Wavelength 220 in.
326
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.
Available land
x
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Section 6-7: Sinusoidal Functions as Mathematical Models
327
Section 6-7: Sinusoidal Functions as Mathematical Models
327
Precalculus with Trigonometry Course Sampler 117
6 -8 Rotary Motion
Sec tion 6 -8
When you ride
rid 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).
PL AN N ING
Rotation
Class Time
EBZT
Farther
(so faster)
Homework Assignment
Day 1: 3" 2o2 1SPCMFNT o PEE
Day 2: 1SPCMFNT o
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Objective
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Important Terms and Concepts
In this exploration you will practice computing and interpreting linear and
angular velocities.
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Exploration Notes
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.
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1. With a flexible ruler you find that the curved
lengths of the arcs are 60 cm (outer) and
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Chapter 6: Applications of Trigonometric and Circular Functions
Precalculus with Trigonometry Course Sampler
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.
continued
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Section Notes
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Section 6-8: Rotary Motion
329
Angle
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Arc Length
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Time
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Linear Velocity
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Outer
___
__
Inner
___
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Section 6-8: Rotary Motion
329
Precalculus with Trigonometry Course Sampler 119
T eac h e R ’ s E dition
EXPLORATION, continued
Differentiating Instruction
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Section 6-8: Rotary Motion
333
Precalculus with Trigonometry Course Sampler
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Chapter 6: Applications of Trigonometric and Circular Functions
Chapter 6: Applications of Trigonometric and Circular Functions
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