Download trigonometry - McDougal Littell

TRIGONOMETRY
SIXTH EDITION
correlated to the
Indiana
Academic Standards for
Precalculus
(Trigonometry Standards 3, 4, 5, 6, 9)
CC2
6/2003
2004
Introduction
to
Trigonometry © 2004
by Roland E. Larson and Robert P. Hostetler
Trigonometry is designed to meet the needs of a trigonometry course covering one semester. The
text introduces a unit-circle approach first and then turns to right triangles. In addition to
trigonometric functions and their graphs, the text covers exponential and logarithmic functions and
analytic geometry (including polar coordinates and parametric equations). Numerous real-life
applications, many using current, real data, are integrated throughout the examples and exercises.
A wide variety of computational, conceptual, and applied problems are graded from less to more
challenging.
Special Features
•
•
•
•
•
Section Openers––section openers include “What you should learn” and “Why you should
learn it,” two features that help students focus while reading and illustrate the relevance of the
section’s content.
P.S. Problem Solving––a set of challenging exercises at the end of each chapter. These
interesting problems not only draw upon and extend the chapter concepts, but they also allude
to concepts that will be discussed in subsequent chapters.
Proofs in Mathematics––this feature emphasizes the importance of proofs in mathematics.
Proofs of important mathematical properties and theorems are presented as well as discussions
of various proof techniques.
Model It––these multi-part applications, referenced in Why you should learn it, offer students
the opportunity to generate and analyze mathematical models.
Algebra of Calculus––special emphasis is given to the algebraic techniques used in calculus.
Algebra of Calculus examples and exercises are integrated throughout the text.
A complete listing of program components is provided on the following page.
i
Trigonometry © 2004
Components
Pupil’s Edition
Instructor’s Annotated Edition
Ancillaries
Student Solutions Guide
Complete Solutions Guide
Test Item File
Student Success Organizer
Technology
HM ClassPrep CD-ROM with HM Testing v6.0
Video/DVD Program
Learning Tools Student CD-ROM
Interactive Trigonometry 3.0 CD-ROM (entire book on CD)
Internet Trigonometry 3.0 (entire book on website)
Textbook web site
ii
Trigonometry © 2004
correlated to
The Indiana Academic Standards for Precalculus (Trigonometry Standards 3, 4, 5, 6, 9)
INSTRUCTION
APPLICATION
Pupil’s Edition and
Teacher’s Edition
Print Ancillaries,
Transparencies and
Technology
STANDARD 3 Trigonometry in Triangles Students define trigonometric functions using right triangles. They
solve word problems and apply the laws of sines and cosines.
PE/IAE
PE/IAE
Test Item File
PC.3.1 Solve word
144-146, 149-150, 196-198
151-154, 163 (#96), 193 (#9194-95
problems involving right
92), 194 (#93-97), 202-205,
Study and Solutions Guide
and oblique triangles.
Ancillaries
214-215
88-90
Example: You want to find
the width of a river that you
cannot cross. You decide to
use a tall tree on the other
bank as a landmark. From a
position directly opposite
the tree, you measure 50 m
along the bank. From that
point, the tree is in a
direction at 37º to your
50 m line. How wide is the
river?
PE/IAE
PC.3.2 Apply the laws of
sines and cosines to solving 274-279, 283-285
problems.
Ancillaries
Example: You want to fix
the location of a mountain
by taking measurements
from two positions 3 miles
apart. From the first
position, the angle between
the mountain and the
second position is 78º. From
the second position, the
angle between the mountain
and the first position is 53º.
How far is the mountain
from each position?
PE/IAE
280-283, 287-290
Test Item File
127-133
Study and Solutions Guide
187-190, 191-194
Learning Tools CD-ROM
Chapter 3: Section 1 Guided
Examples 1, 2, 3, 5, 6
PE/IAE
280, 282 (#48)
Test Item File
127-129
Study and Solutions Guide
189
Learning Tools CD-ROM
Chapter 3, Section 1, Guided
Example 4
Study and Solutions Guide
187-190
Learning Tools CD-ROM
Chapter 3: Section 1,
Section 2 Concept
Student Success Organizer
76
PC.3.3 Find the area of a PE/IAE
triangle given two sides and 278-279
the angle between them.
Ancillaries
Example: Calculate the
area of a triangle with sides
of length 8 cm and 6 cm
enclosing an angle of 60º.
Learning Tools CD-ROM
Chapter 1: Section 1.3 Guided
Examples 1, 7
Study and Solutions Guide
88
Learning Tools CD-ROM
Chapter 1: Section 1.3
Concept
Student Success Organizer
61
Study and Solutions Guide
187
PE = Pupil’s Edition, IAE = Instructor’s Annotated Edition
Selected exercises are referenced in parentheses, otherwise entire page is applicable.
1
Trigonometry © 2004 correlated to the
Indiana Academic Standards for Precalculus (Trigonometry Standards 3, 4, 5, 6, 9)
INSTRUCTION
APPLICATION
Pupil’s Edition and
Teacher’s Edition
Print Ancillaries,
Transparencies and
Technology
STANDARD 4 Trigonometric Functions Students define trigonometric functions using the unit circle and use
degrees and radians. They draw and analyze graphs, find inverse functions, and solve word problems.
PE/IAE
PE/IAE
Study and Solutions Guide
PC.4.1 Define sine and
142, 161 (#37-42), 162
86, 97, & 98
cosine using the unit circle. 137, 138, 157-159
Learning Tools CD-ROM
Ancillaries
Study and Solutions Guide
85 & 94
Learning Tools CD-ROM
Chapter 1, Section 1.2
Concept: Unit Circle
(Animation)
Student Success Organizer
40
Example: Find the acute
angle A for which
sin 150º = sin A.
PE/IAE
130-131
PC.4.2 Convert between
degree and radian
measures.
IAE Only:
133-134
Study and Solutions Guide
82
Learning Tools CD-ROM
Chapter 1, Section 1.1,
Guided Examples 6, 7
PE/IAE
151, 161 (#29-36), 162
Test Item File
77-79
Study and Solutions Guide
91, 97
Learning Tools CD-ROM
Chapter 1, Section 1.3,
Guided Example 2
Ancillaries
Study and Solutions Guide
80
Student Success Organizer
37
Example: Convert 90º, 45º,
and 30º to radians.
PC.4.3 Learn exact sine,
cosine, and tangent values
for 0, π/2, π/3, π/4, π/6, and
multiples of π.
Use those values to find
other trigonometric values.
Example: Find the values of
cos π/2, tan3π/4, csc2π/3,
sin-1 – √3/2 and sin 3π.
Chapter 1, Section 1.4,
Guided Example 5
PE/IAE
145-146, 158-159
Ancillaries
Study and Solutions Guide
88, 94
Learning Tools CD-ROM
Chapter 1, Section 1.3,
Animation
Student Success Organizer
44
PE = Pupil’s Edition, IAE = Instructor’s Annotated Edition
Selected exercises are referenced in parentheses, otherwise entire page is applicable.
2
Trigonometry © 2004 correlated to the
Indiana Academic Standards for Precalculus (Trigonometry Standards 3, 4, 5, 6, 9)
INSTRUCTION
APPLICATION
Pupil’s Edition and
Teacher’s Edition
PE/IAE
170, 199-201
PC.4.4 Solve word
problems involving
applications of
trigonometric functions.
PE/IAE
164-170, 175-179
PC.4.5 Define and graph
trigonometric functions
(i.e., sine, cosine, tangent,
cosecant, secant,
cotangent).
PC.4.6 Find domain,
range, intercepts, periods,
amplitudes, and asymptotes
of trigonometric functions.
PC.4.7 Draw and analyze
graphs of translations of
trigonometric functions,
including period,
amplitude, and phase shift.
Test Item File
93-95
Study and Solutions Guide
104, 105, 110-111
Learning Tools CD-ROM
Chapter 1, Section 1.3,
Guided Example 5
PE/IAE
171-173, 182-184
Test Item File
85-91
Study and Solutions Guide
100-104, 106-108
Learning Tools CD-ROM
All Guided Examples
PE/IAE
171, 182
Test Item File
85-91
Study and Solutions Guide
100-101, 106
HM ClassPrep CD-ROM
Chapter 1, Section 1.5,
Guided Example 1;
Chapter 1, Section 1.6,
Guided Examples 1-4
PE/IAE
171-172, 182
Test Item File
79-84
Study and Solutions Guide
101, 102, 108
Learning Tools CD-ROM
Chapter 1, Section 1.5,
Editable Graph Exploration,
Guided Examples 2-3
Ancillaries
Study and Solutions Guide
100-106
Learning Tools CD-ROM
Chapter 1, Section 1.5,
Concepts and Animations
Student Success Organizer
52
Example: Graph y = sin x
and y = cos x, and compare
their graphs.
Example: Find the
asymptotes of tan x and find
its domain.
PE/IAE
172-174, 183 (#73-74), 184,
205-206
Ancillaries
Learning Tools CD-ROM
Chapter 1, Section 1.3,
Concept: Mathematical
Modeling
Example: In Indiana, the
day length in hours varies
through the year in a sine
wave. The longest day of 14
hours is on Day 175 and the
shortest day of 10 hours is
on Day 355. Sketch a graph
of this function and find its
formula. Which other day
has the same length as July
4?
PE/IAE
164-167, 175-179
Ancillaries
Study and Solutions Guide
100-106
Learning Tools CD-ROM
Chapter 1, Section 1.5,
Concept: Amplitude and
Period
Student Success Organizer
53
PE/IAE
168-169
Print Ancillaries,
Transparencies and
Technology
Ancillaries
Study and Solutions Guide
100, 106
Example: Draw the graph of
y = 5 + sin (x – π/3).
PE = Pupil’s Edition, IAE = Instructor’s Annotated Edition
Selected exercises are referenced in parentheses, otherwise entire page is applicable.
3
Trigonometry © 2004 correlated to the
Indiana Academic Standards for Precalculus (Trigonometry Standards 3, 4, 5, 6, 9)
INSTRUCTION
APPLICATION
Pupil’s Edition and
Teacher’s Edition
PC.4.8 Define and graph
inverse trigonometric
functions.
Example: Graph
f(x) = sin-1x.
PC.4.9 Find values of
trigonometric and inverse
trigonometric functions.
PE/IAE
186-189
PE/IAE
193, 195 (#105-106)
Test Item File
91-93
Study and Solutions Guide
115
Learning Tools CD-ROM
Chapter 1, Section 1.7,
Synthesis Example 1
PE/IAE
142, 151-152, 161-162, 192
Test Item File
70-79, 91-93
Study and Solutions Guide
112-113
Learning Tools CD-ROM
Chapter 1, Section 1.7,
Guided Example 1
PE/IAE
430
Study and Solutions Guide
284 & 285
Learning Tools CD-ROM
Chapter 6, Section 6.1,
Guided Examples 1-2
PE/IAE
161-162
Test Item File
74-79
Learning Tools CD-ROM
Chapter 1, Section 1.3,
Guided Example 2
Ancillaries
Study and Solutions Guide
112
Learning Tools CD-ROM
Chapter 1, Section 1.7,
Graphing the Arcsine
Function and Other Inverse
Trig. Functions (Animation)
Student Success Organizer
54
PE/IAE
138-140, 144-146, 155-159,
187, 189
Example: Find the values of
sin π/2 and tan-1 √3.
PE/IAE
PC.4.10 Know that the
tangent of the angle that a 426-427, 505
line makes with the x-axis is Ancillaries
equal to the slope of the
Study and Solutions Guide
line.
284
Example: Use a right
triangle to show that the
slope of a line at 135º to the
x-axis is -1.
PC.4.11 Make connections
between right triangle
ratios, trigonometric
functions, and circular
functions.
Example: Angle A is a 60º
angle of a right triangle
with a hypotenuse of length
14 and a shortest side of
length 7. Find the exact
sine, cosine, and tangent of
angle A. Find the realπ
numbers x, 0 < x < 2 ,
with exactly the same sine,
cosine, and tangent values.
Print Ancillaries,
Transparencies and
Technology
Learning Tools CD-ROM
Chapter 6, Section 6.1,
Concept: Inclination of a
Line, Simulation: Finding the
slope and inclination of a
line.
PE/IAE
144-146, 155-159
Ancillaries
Learning Tools CD-ROM
Chapter 1, Section 1.3,
Animation: Finding Sines,
Cosines, and Tangents of
Special Angles; Section 1.4,
Synthesis Example 1
PE = Pupil’s Edition, IAE = Instructor’s Annotated Edition
Selected exercises are referenced in parentheses, otherwise entire page is applicable.
4
Trigonometry © 2004 correlated to the
Indiana Academic Standards for Precalculus (Trigonometry Standards 3, 4, 5, 6, 9)
INSTRUCTION
APPLICATION
Pupil’s Edition and
Teacher’s Edition
Print Ancillaries,
Transparencies and
Technology
STANDARD 5 Trigonometric Identities and Equations Students prove trigonometric identities, solve
trigonometric equations, and solve word problems.
PE/IAE
PE/IAE
PC.5.1 Know the basic
147, 218-219
225 (#113)
trigonometric identity
2
2
cos x + sin x = 1 and prove Ancillaries
that it is equivalent to the
Study and Solutions Guide
Pythagorean Theorem.
133
Example: Use a right
triangle to show that
cos2x + sin2x = 1.
PC.5.2 Use basic
trigonometric identities to
verify other identities and
simplify expressions.
Example: Show that
(tan2x)/(1+tan2x) = sin2 x.
PC.5.3 Understand and
use the addition formulas
for sines, cosines, and
tangents.
Example: Prove that
sin (A + B) = sinA cosB +
cosA sinB and use it to find
a formula for sin 2x.
Student Success Organizer
63
PE/IAE
226-230, 218-220
PE/IAE
223-225, 231-232
Test Item File
111
Study and Solutions Guide
135, 140-142
Learning Tools CD-ROM
Chapter 2, Section 2.2,
Guided Examples 1-3;
Chapter 2, Section 2.1,
Guided Example 5
PE/IAE
248-250
Test Item File
115-116
Study and Solutions Guide
152-156
Ancillaries
Study and Solutions Guide
140 & 133
Learning Tools CD-ROM
Chapter 2, Section 2.2,
Concept: Introduction,
Animation: Verifying a
Trigonometric Identity
Student Success Organizer
65-66
PE/IAE
244-247, 267-269
Ancillaries
Study and Solutions Guide
152
Learning Tools CD-ROM
Chapter 2, Section 2.4,
Concept: Using Sum and
Difference Formulas
(Synthesis Example 1)
Student Success Organizer
69
PE = Pupil’s Edition, IAE = Instructor’s Annotated Edition
Selected exercises are referenced in parentheses, otherwise entire page is applicable.
5
Trigonometry © 2004 correlated to the
Indiana Academic Standards for Precalculus (Trigonometry Standards 3, 4, 5, 6, 9)
INSTRUCTION
APPLICATION
Pupil’s Edition and
Teacher’s Edition
PC.5.4 Understand and
use the half-angle and
double-angle formulas for
sines, cosines, and tangents.
PE/IAE
251-252, 254-255
Ancillaries
Study and Solutions Guide
162
Learning Tools CD-ROM
Example: Prove that
Chapter 2, Section 2.5,
2
+
cos x = 1/2 1/2 (cos2x).
Concept: Half-Angle
Formulas, Animation:
Deriving the Half-angle
Formulas
Student Success Organizer
71-72
PC.5.5 Solve trigonometric PE/IAE
233-235
equations.
Example:
Solve 3 sin 2x = 1 for x
between 0 and 2π.
PC.5.6 Solve word
problems involving
applications of
trigonometric equations.
PE/IAE
258-259
Test Item File
116-119
Study and Solutions Guide
162-169
Learning Tools CD-ROM
Chapter 2, Section 2.5,
Guided Example 1
PE/IAE
240-241, 249 (#73-74), 258 (#918), 259 (#59-62, 87-90)
Test Item File
113-115
Study and Solutions Guide
145-148
Learning Tools CD-ROM
Chapter 2, Section 2.3,
Guided Exercises 1-7
PE/IAE
242-243; 270-271
Study and Solutions Guide
150-151, 183-184
Ancillaries
Study and Solutions Guide
144
Learning Tools CD-ROM
Chapter 2, Section 2.3,
Synthesis Examples 1-2
Student Success Organizer
67, 70
Ancillaries
Learning Tools CD-ROM
Chapter 2, Section 2.3,
Synthesis Example 2
Print Ancillaries,
Transparencies and
Technology
Example: In the example
about day length in
Standard 4, for how long in
winter is there less than 11
hours of daylight?
PE = Pupil’s Edition, IAE = Instructor’s Annotated Edition
Selected exercises are referenced in parentheses, otherwise entire page is applicable.
6
Trigonometry © 2004 correlated to the
Indiana Academic Standards for Precalculus (Trigonometry Standards 3, 4, 5, 6, 9)
INSTRUCTION
APPLICATION
Pupil’s Edition and
Teacher’s Edition
Print Ancillaries,
Transparencies and
Technology
STANDARD 6 Polar Coordinates and Complex Numbers Students define polar coordinates and complex
numbers and understand their connection with trigonometric functions.
PE/IAE
PE/IAE
Test Item File
PC.6.1 Define polar
476-478
480
206-208
coordinates and relate
Study and Solutions Guide
polar coordinates to
Ancillaries
320-321
Cartesian coordinates.
Learning Tools CD-ROM
Study and Solutions Guide
Example: Convert the polar
coordinates (2, π/3) to
(x, y) form.
PC.6.2 Represent
equations given in
rectangular coordinates in
terms of polar coordinates.
Example: Represent the
equation x2 + y2 = 4 in
terms of polar coordinates.
PE/IAE
479
PC.6.4 Define complex
numbers, convert complex
numbers to trigonometric
form, and multiply
complex numbers in
trigonometric form.
Example: Write 3 + 3i and
2 – 4i in trigonometric form
and then multiply the
results.
IAE Only
480
Study and Solutions Guide
321-324
Learning Tools CD-ROM
Chapter 6, Section 6.7,
Guided Examples 3-6
PE/IAE
488-489
Test Item File
208-214
Study and Solutions Guide
325-330
Learning Tools CD-ROM
Chapter 6, Section 6.8,
Guided Exercises 1-4
PE/IAE
333-334, 347-348
Test Item File
149-150
Study and Solutions Guide
224-228
Learning Tools CD-ROM
Chapter 4, Section 4.3,
Guided Examples 2, 4
Ancillaries
Study and Solutions Guide
320
Learning Tools CD-ROM
Student Success Organizer
124
PC.6.3 Graph equations in PE/IAE
the polar coordinate plane. 482-487
Example: Graph
y = 1 – cos Θ
Chapter 6, Section 6.7,
Guided Examples 1-2
320
Learning Tools CD-ROM
Chapter 6, Section 6.7,
Animation: Plotting Points in
Rectangular and Polar
Coordinate Systems
Student Success Organizer
123-124
Ancillaries
Study and Solutions Guide
325
Learning Tools CD-ROM
Chapter 6, Section 6.8,
Synthesis Examples 1-3;
Animation: Sketching a Rose
Curve
Student Success Organizer
125-126
PE/IAE
328-331, 342-346
Ancillaries
Study and Solutions Guide
223
Learning Tools CD-ROM
Chapter 4, Section 4.3,
Synthesis Example 2
Student Success Organizer
93-94
PE = Pupil’s Edition, IAE = Instructor’s Annotated Edition
Selected exercises are referenced in parentheses, otherwise entire page is applicable.
7
Trigonometry © 2004 correlated to the
Indiana Academic Standards for Precalculus (Trigonometry Standards 3, 4, 5, 6, 9)
INSTRUCTION
APPLICATION
Pupil’s Edition and
Teacher’s Edition
PE/IAE
PC.6.5 State, prove, and
use De Moivre’s Theorem. 349-352
Example:
Simplify (1 – i)23.
PE/IAE:
353-354
Ancillaries
Study and Solutions Guide
229
Student Success Organizer
95
Print Ancillaries,
Transparencies and
Technology
Test Item File
155-156
Study and Solutions Guide
230-231
Learning Tools CD-ROM
Chapter 4, Section 4.4,
Guided Examples 1, 2
PE = Pupil’s Edition, IAE = Instructor’s Annotated Edition
Selected exercises are referenced in parentheses, otherwise entire page is applicable.
8
Trigonometry © 2004 correlated to the
Indiana Academic Standards for Precalculus (Trigonometry Standards 3, 4, 5, 6, 9)
INSTRUCTION
APPLICATION
Pupil’s Edition and
Teacher’s Edition
Print Ancillaries,
Transparencies and
Technology
STANDARD 9 Mathematical Reasoning and Problem Solving Students use a variety of strategies to solve
problems.
Found throughout the text.
PC.9.1 Use a variety of
See, for example:
problem-solving strategies,
such as drawing a diagram,
PE/IAE
guess-and-check, solving a
122, 214, 270, 324, 360, 422,
simpler problem,
508
examining simpler
problems, and working
backwards.
Example: The half-life of
carbon-14 is 5,730 years.
The original concentration
of carbon-14 in a living
organism was 500 grams.
How might you find the age
of a fossil of that living
organism with a carbon-14
concentration of 140 grams?
PC.9.2 Decide whether a
solution is reasonable in the
context of the original
situation.
Example: John says the
answer to the problem in
the first example is about
10,000 years. Is his answer
reasonable? Why or why
not?
Opportunities for students to
check the reasonableness of
their results are found
throughout the text. See, for
example:
PE/IAE
122, 214, 230, 270, 324, 360,
422, 508
PE = Pupil’s Edition, IAE = Instructor’s Annotated Edition
Selected exercises are referenced in parentheses, otherwise entire page is applicable.
9
Trigonometry © 2004 correlated to the
Indiana Academic Standards for Precalculus (Trigonometry Standards 3, 4, 5, 6, 9)
INSTRUCTION
APPLICATION
Pupil’s Edition and
Teacher’s Edition
Print Ancillaries,
Transparencies and
Technology
Students develop and evaluate mathematical arguments and proofs.
Opportunities to address this
standard can be found
throughout the text. See, for
example:
PC.9.3 Decide if a given
algebraic statement is true
always, sometimes, or
never (statements involving
rational or radical
expressions, trigonometric,
logarithmic or exponential
functions).
PE/IAE:
11 (#111-112), 66 (#100-101),
136 (#103-105), 210 (#133-136),
232 (#59-60), 334 (#85-87),
374 (#61-62), 441 (#69-70),
496 (#59-60), 501 (#109-112)
Example: Is the statement
sin 2x = 2 sinx cosx true
always, sometimes, or
never? Explain your answer.
PC.9.4 Use the properties
of number systems and
order of operations to
justify the steps of
simplifying functions and
solving equations.
PE/IAE:
13-20, 226-230, 233-239
PE/IAE:
21-23, 231-232, 240-243
Test Item File
3-6
Example: Simplify
5
(
+
x −2
1
x +3
2
x +3
+
)
÷
7
x−2
(
),
explaining why you can
take each step.
PE = Pupil’s Edition, IAE = Instructor’s Annotated Edition
Selected exercises are referenced in parentheses, otherwise entire page is applicable.
10
Trigonometry © 2004 correlated to the
Indiana Academic Standards for Precalculus (Trigonometry Standards 3, 4, 5, 6, 9)
INSTRUCTION
APPLICATION
Pupil’s Edition and
Teacher’s Edition
PC.9.5 Understand that the PE/IAE
19, 236
logic of equation solving
begins with the assumption
that the variable is a
number that satisfies the
equation, and that the steps
taken when solving
equations create new
equations that have, in most
cases, the same solution set
as the original. Understand
that similar logic applies to
solving systems of equations
simultaneously.
Example: A student solving
the equation
x + √x–30 = 0 comes up
with the solution set
{25, 36}. Explain why
{25, 36} is not the solution
set to this equation, and
why the “check” step is
essential in solving the
equation.
PC.9.6 Define and use the
mathematical induction
method of proof.
Example: Prove
De Moivre’s Theorem using
mathematical induction.
Print Ancillaries,
Transparencies and
Technology
PE/IAE
22, 23, 136, 154
A variety of proofs are
presented throughout the text
in the “Proofs in
Mathematics” feature. See:
PE/IAE
121, 213, 267, 320, 359, 421,
505
PE = Pupil’s Edition, IAE = Instructor’s Annotated Edition
Selected exercises are referenced in parentheses, otherwise entire page is applicable.
11