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

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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

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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

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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.

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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

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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

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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

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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

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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

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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.

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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