TRIGONOMETRY - McDougal · PDF fileTrigonometry is designed to meet the needs of a trigonometry course covering one semester. The text introduces a unit-circle approach first and then

IndianaAcademic Standards for

Precalculus(Trigonometry Standards 3, 4, 5, 6, 9)

correlated to the

6/20032004

CC2

TRIGONOMETRYS I X T H E D I T I O N

i

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. Thetext introduces a unit-circle approach first and then turns to right triangles. In addition totrigonometric functions and their graphs, the text covers exponential and logarithmic functions andanalytic geometry (including polar coordinates and parametric equations). Numerous real-lifeapplications, 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 morechallenging.

Special Features

• Section Openers––section openers include “What you should learn” and “Why you shouldlearn it,” two features that help students focus while reading and illustrate the relevance of thesection’s content.

• P.S. Problem Solving––a set of challenging exercises at the end of each chapter. Theseinteresting problems not only draw upon and extend the chapter concepts, but they also alludeto 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 discussionsof various proof techniques.

• Model It––these multi-part applications, referenced in Why you should learn it, offer studentsthe 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.

ii

Trigonometry © 2004Components

Pupil’s EditionInstructor’s Annotated Edition

AncillariesStudent Solutions GuideComplete Solutions GuideTest Item FileStudent Success Organizer

Technology HM ClassPrep CD-ROM with HM Testing v6.0Video/DVD ProgramLearning Tools Student CD-ROMInteractive Trigonometry 3.0 CD-ROM (entire book on CD)Internet Trigonometry 3.0 (entire book on website)Textbook web site

PE = Pupil’s Edition, IAE = Instructor’s Annotated Edition

Selected exercises are referenced in parentheses, otherwise entire page is applicable.1

Trigonometry © 2004correlated to

The Indiana Academic Standards for Precalculus (Trigonometry Standards 3, 4, 5, 6, 9)

INSTRUCTION APPLICATION

Pupil’s Edition andTeacher’s Edition

Print Ancillaries,Transparencies andTechnology

STANDARD 3 Trigonometry in Triangles Students define trigonometric functions using right triangles. Theysolve word problems and apply the laws of sines and cosines.PC.3.1 Solve wordproblems involving rightand oblique triangles.

Example: You want to findthe width of a river that youcannot cross. You decide touse a tall tree on the otherbank as a landmark. From aposition directly oppositethe tree, you measure 50 malong the bank. From thatpoint, the tree is in adirection at 37º to your50 m line. How wide is theriver?

PE/IAE144-146, 149-150, 196-198

Ancillaries Study and Solutions Guide88Learning Tools CD-ROMChapter 1: Section 1.3ConceptStudent Success Organizer61

PE/IAE151-154, 163 (#96), 193 (#91-92), 194 (#93-97), 202-205,214-215

Test Item File94-95Study and Solutions Guide88-90Learning Tools CD-ROMChapter 1: Section 1.3 GuidedExamples 1, 7

PC.3.2 Apply the laws ofsines and cosines to solvingproblems.

Example: You want to fixthe location of a mountainby taking measurementsfrom two positions 3 milesapart. From the firstposition, the angle betweenthe mountain and thesecond position is 78º. Fromthe second position, theangle between the mountainand the first position is 53º.How far is the mountainfrom each position?

PE/IAE274-279, 283-285

Ancillaries Study and Solutions Guide187-190Learning Tools CD-ROMChapter 3: Section 1,Section 2 ConceptStudent Success Organizer76

PE/IAE280-283, 287-290

Test Item File127-133Study and Solutions Guide187-190, 191-194Learning Tools CD-ROMChapter 3: Section 1 GuidedExamples 1, 2, 3, 5, 6

PC.3.3 Find the area of atriangle given two sides andthe angle between them.

Example: Calculate thearea of a triangle with sidesof length 8 cm and 6 cmenclosing an angle of 60º.

PE/IAE278-279

Ancillaries Study and Solutions Guide187

PE/IAE280, 282 (#48)

Test Item File127-129Study and Solutions Guide189Learning Tools CD-ROMChapter 3, Section 1, GuidedExample 4

Trigonometry © 2004 correlated to theIndiana Academic Standards for Precalculus (Trigonometry Standards 3, 4, 5, 6, 9)






STANDARD 4 Trigonometric Functions Students define trigonometric functions using the unit circle and usedegrees and radians. They draw and analyze graphs, find inverse functions, and solve word problems.

PC.4.1 Define sine andcosine using the unit circle.

Example: Find the acuteangle A for whichsin 150º = sin A.

PE/IAE137, 138, 157-159

Ancillaries Study and Solutions Guide85 & 94Learning Tools CD-ROMChapter 1, Section 1.2Concept: Unit Circle(Animation)Student Success Organizer40

PE/IAE142, 161 (#37-42), 162

Study and Solutions Guide86, 97, & 98Learning Tools CD-ROMChapter 1, Section 1.4,Guided Example 5

PC.4.2 Convert betweendegree and radianmeasures.

Example: Convert 90º, 45º,and 30º to radians.

PE/IAE130-131

Ancillaries Study and Solutions Guide80Student Success Organizer37

IAE Only:133-134

Study and Solutions Guide82Learning Tools CD-ROMChapter 1, Section 1.1,Guided Examples 6, 7

PC.4.3 Learn exact sine,cosine, and tangent valuesfor 0, π/2, π/3, π/4, π/6, andmultiples of π.Use those values to findother trigonometric values.

Example: Find the values ofcos π/2, tan3π/4, csc2π/3,sin-1 – √3/2 and sin 3π.

PE/IAE145-146, 158-159

Ancillaries Study and Solutions Guide88, 94Learning Tools CD-ROMChapter 1, Section 1.3,AnimationStudent Success Organizer44

PE/IAE151, 161 (#29-36), 162

Test Item File77-79Study and Solutions Guide91, 97Learning Tools CD-ROMChapter 1, Section 1.3,Guided Example 2







PC.4.4 Solve wordproblems involvingapplications oftrigonometric functions.

Example: In Indiana, theday length in hours variesthrough the year in a sinewave. The longest day of 14hours is on Day 175 and theshortest day of 10 hours ison Day 355. Sketch a graphof this function and find itsformula. Which other dayhas the same length as July4?

PE/IAE170, 199-201

Ancillaries Learning Tools CD-ROMChapter 1, Section 1.3,Concept: MathematicalModeling

PE/IAE172-174, 183 (#73-74), 184,205-206

Test Item File93-95Study and Solutions Guide104, 105, 110-111Learning Tools CD-ROMChapter 1, Section 1.3,Guided Example 5

PC.4.5 Define and graphtrigonometric functions(i.e., sine, cosine, tangent,cosecant, secant,cotangent).

Example: Graph y = sin xand y = cos x, and comparetheir graphs.

PE/IAE164-170, 175-179

Ancillaries Study and Solutions Guide100-106Learning Tools CD-ROMChapter 1, Section 1.5,Concepts and AnimationsStudent Success Organizer52

PE/IAE171-173, 182-184

Test Item File85-91Study and Solutions Guide100-104, 106-108Learning Tools CD-ROMAll Guided Examples

PC.4.6 Find domain,range, intercepts, periods,amplitudes, and asymptotesof trigonometric functions.

Example: Find theasymptotes of tan x and findits domain.

PE/IAE164-167, 175-179

Ancillaries Study and Solutions Guide100-106Learning Tools CD-ROMChapter 1, Section 1.5,Concept: Amplitude andPeriodStudent Success Organizer53

PE/IAE171, 182

Test Item File85-91Study and Solutions Guide100-101, 106HM ClassPrep CD-ROMChapter 1, Section 1.5,Guided Example 1;Chapter 1, Section 1.6,Guided Examples 1-4

PC.4.7 Draw and analyzegraphs of translations oftrigonometric functions,including period,amplitude, and phase shift.

Example: Draw the graph ofy = 5 + sin (x – π/3).

PE/IAE168-169

Ancillaries Study and Solutions Guide100, 106

PE/IAE171-172, 182

Test Item File79-84Study and Solutions Guide101, 102, 108Learning Tools CD-ROMChapter 1, Section 1.5,Editable Graph Exploration,Guided Examples 2-3







PC.4.8 Define and graphinverse trigonometricfunctions.

Example: Graphf(x) = sin-1x.

PE/IAE186-189

Ancillaries Study and Solutions Guide112Learning Tools CD-ROMChapter 1, Section 1.7,Graphing the ArcsineFunction and Other InverseTrig. Functions (Animation)Student Success Organizer54

PE/IAE193, 195 (#105-106)

Test Item File91-93Study and Solutions Guide115Learning Tools CD-ROMChapter 1, Section 1.7,Synthesis Example 1

PC.4.9 Find values oftrigonometric and inversetrigonometric functions.

Example: Find the values ofsin π/2 and tan-1 √3.

PE/IAE138-140, 144-146, 155-159,187, 189

PE/IAE

142, 151-152, 161-162, 192

Test Item File70-79, 91-93Study and Solutions Guide112-113Learning Tools CD-ROMChapter 1, Section 1.7,Guided Example 1

PC.4.10 Know that thetangent of the angle that aline makes with the x-axis isequal to the slope of theline.

Example: Use a righttriangle to show that theslope of a line at 135º to thex-axis is -1.

PE/IAE426-427, 505

Ancillaries Study and Solutions Guide284Learning Tools CD-ROMChapter 6, Section 6.1,Concept: Inclination of aLine, Simulation: Finding theslope and inclination of aline.

PE/IAE430

Study and Solutions Guide284 & 285Learning Tools CD-ROMChapter 6, Section 6.1,Guided Examples 1-2

PC.4.11 Make connectionsbetween right triangleratios, trigonometricfunctions, and circularfunctions.

Example: Angle A is a 60ºangle of a right trianglewith a hypotenuse of length14 and a shortest side oflength 7. Find the exactsine, cosine, and tangent ofangle A. Find the realnumbers x, 0 < x < 2

π,

with exactly the same sine,cosine, and tangent values.

PE/IAE144-146, 155-159

Ancillaries Learning Tools CD-ROMChapter 1, Section 1.3,Animation: Finding Sines,Cosines, and Tangents ofSpecial Angles; Section 1.4,Synthesis Example 1

PE/IAE161-162

Test Item File74-79Learning Tools CD-ROMChapter 1, Section 1.3,Guided Example 2







STANDARD 5 Trigonometric Identities and Equations Students prove trigonometric identities, solvetrigonometric equations, and solve word problems.PC.5.1 Know the basictrigonometric identitycos2x + sin2x = 1 and provethat it is equivalent to thePythagorean Theorem.

Example: Use a righttriangle to show thatcos2x + sin2x = 1.

PE/IAE147, 218-219


PE/IAE225 (#113)

PC.5.2 Use basictrigonometric identities toverify other identities andsimplify expressions.

Example: Show that(tan2x)/(1+tan2x) = sin2 x.

PE/IAE226-230, 218-220

Ancillaries Study and Solutions Guide140 & 133Learning Tools CD-ROMChapter 2, Section 2.2,Concept: Introduction,Animation: Verifying aTrigonometric IdentityStudent Success Organizer65-66

PE/IAE223-225, 231-232

Test Item File111Study and Solutions Guide135, 140-142Learning Tools CD-ROMChapter 2, Section 2.2,Guided Examples 1-3;Chapter 2, Section 2.1,Guided Example 5

PC.5.3 Understand anduse the addition formulasfor sines, cosines, andtangents.

Example: Prove thatsin (A + B) = sinA cosB +cosA sinB and use it to finda formula for sin 2x.

PE/IAE244-247, 267-269

Ancillaries Study and Solutions Guide152Learning Tools CD-ROMChapter 2, Section 2.4,Concept: Using Sum andDifference Formulas(Synthesis Example 1)Student Success Organizer69

PE/IAE248-250

Test Item File115-116Study and Solutions Guide152-156







PC.5.4 Understand anduse the half-angle anddouble-angle formulas forsines, cosines, and tangents.

Example: Prove that

cos2x = 1/2 + 1/2 (cos2x).

PE/IAE251-252, 254-255

Ancillaries Study and Solutions Guide162Learning Tools CD-ROMChapter 2, Section 2.5,Concept: Half-AngleFormulas, Animation:Deriving the Half-angleFormulasStudent Success Organizer71-72

PE/IAE258-259

Test Item File116-119Study and Solutions Guide162-169Learning Tools CD-ROMChapter 2, Section 2.5,Guided Example 1

PC.5.5 Solve trigonometricequations.

Example:Solve 3 sin 2x = 1 for xbetween 0 and 2π.

PE/IAE233-235

Ancillaries Study and Solutions Guide144Learning Tools CD-ROMChapter 2, Section 2.3,Synthesis Examples 1-2Student Success Organizer67, 70

PE/IAE240-241, 249 (#73-74), 258 (#9-18), 259 (#59-62, 87-90)

Test Item File113-115Study and Solutions Guide145-148Learning Tools CD-ROMChapter 2, Section 2.3,Guided Exercises 1-7

PC.5.6 Solve wordproblems involvingapplications oftrigonometric equations.

Example: In the exampleabout day length inStandard 4, for how long inwinter is there less than 11hours of daylight?

Ancillaries Learning Tools CD-ROMChapter 2, Section 2.3,Synthesis Example 2

PE/IAE242-243; 270-271

Study and Solutions Guide150-151, 183-184







STANDARD 6 Polar Coordinates and Complex Numbers Students define polar coordinates and complexnumbers and understand their connection with trigonometric functions.PC.6.1 Define polarcoordinates and relatepolar coordinates toCartesian coordinates.

Example: Convert the polarcoordinates (2, π/3) to(x, y) form.

PE/IAE476-478

Ancillaries Study and Solutions Guide320Learning Tools CD-ROMChapter 6, Section 6.7,Animation: Plotting Points inRectangular and PolarCoordinate SystemsStudent Success Organizer123-124

PE/IAE480

Test Item File206-208Study and Solutions Guide320-321Learning Tools CD-ROMChapter 6, Section 6.7,Guided Examples 1-2

PC.6.2 Representequations given inrectangular coordinates interms of polar coordinates.

Example: Represent theequation x2 + y2 = 4 interms of polar coordinates.

PE/IAE479

Ancillaries Study and Solutions Guide320Learning Tools CD-ROMStudent Success Organizer124

IAE Only480

Study and Solutions Guide321-324Learning Tools CD-ROMChapter 6, Section 6.7,Guided Examples 3-6

PC.6.3 Graph equations inthe polar coordinate plane.

Example: Graphy = 1 – cos Θ

PE/IAE482-487

Ancillaries Study and Solutions Guide325Learning Tools CD-ROMChapter 6, Section 6.8,Synthesis Examples 1-3;Animation: Sketching a RoseCurveStudent Success Organizer125-126

PE/IAE488-489

Test Item File208-214Study and Solutions Guide325-330Learning Tools CD-ROMChapter 6, Section 6.8,Guided Exercises 1-4

PC.6.4 Define complexnumbers, convert complexnumbers to trigonometricform, and multiplycomplex numbers intrigonometric form.

Example: Write 3 + 3i and2 – 4i in trigonometric formand then multiply theresults.

PE/IAE328-331, 342-346

Ancillaries Study and Solutions Guide223Learning Tools CD-ROMChapter 4, Section 4.3,Synthesis Example 2Student Success Organizer93-94

PE/IAE333-334, 347-348

Test Item File149-150Study and Solutions Guide224-228Learning Tools CD-ROMChapter 4, Section 4.3,Guided Examples 2, 4







PC.6.5 State, prove, anduse De Moivre’s Theorem.

Example:Simplify (1 – i)23.

PE/IAE349-352


PE/IAE:353-354

Test Item File155-156Study and Solutions Guide230-231Learning Tools CD-ROMChapter 4, Section 4.4,Guided Examples 1, 2







STANDARD 9 Mathematical Reasoning and Problem Solving Students use a variety of strategies to solveproblems.PC.9.1 Use a variety ofproblem-solving strategies,such as drawing a diagram,guess-and-check, solving asimpler problem,examining simplerproblems, and workingbackwards.

Example: The half-life ofcarbon-14 is 5,730 years.The original concentrationof carbon-14 in a livingorganism was 500 grams.How might you find the ageof a fossil of that livingorganism with a carbon-14concentration of 140 grams?

Found throughout the text.See, for example:

PE/IAE122, 214, 270, 324, 360, 422,508

PC.9.2 Decide whether asolution is reasonable in thecontext of the originalsituation.

Example: John says theanswer to the problem inthe first example is about10,000 years. Is his answerreasonable? Why or whynot?

Opportunities for students tocheck the reasonableness oftheir results are foundthroughout the text. See, forexample:

PE/IAE122, 214, 230, 270, 324, 360,422, 508







Students develop and evaluate mathematical arguments and proofs.

PC.9.3 Decide if a givenalgebraic statement is truealways, sometimes, ornever (statements involvingrational or radicalexpressions, trigonometric,logarithmic or exponentialfunctions).

Example: Is the statementsin 2x = 2 sinx cosx truealways, sometimes, ornever? Explain your answer.

Opportunities to address thisstandard can be foundthroughout the text. See, forexample:

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)

PC.9.4 Use the propertiesof number systems andorder of operations tojustify the steps ofsimplifying functions andsolving equations.

Example: Simplify

(3

2

2

5

++

− xx) ÷

(2

7

3

1

−+

+ xx) ,

explaining why you cantake each step.

PE/IAE:13-20, 226-230, 233-239

PE/IAE:21-23, 231-232, 240-243

Test Item File3-6







PC.9.5 Understand that thelogic of equation solvingbegins with the assumptionthat the variable is anumber that satisfies theequation, and that the stepstaken when solvingequations create newequations that have, in mostcases, the same solution setas the original. Understandthat similar logic applies tosolving systems of equationssimultaneously.

Example: A student solvingthe equationx + √x–30 = 0 comes upwith the solution set{25, 36}. Explain why{25, 36} is not the solutionset to this equation, andwhy the “check” step isessential in solving theequation.

PE/IAE19, 236

PE/IAE22, 23, 136, 154

PC.9.6 Define and use themathematical inductionmethod of proof.

Example: ProveDe Moivre’s Theorem usingmathematical induction.

A variety of proofs arepresented throughout the textin the “Proofs inMathematics” feature. See:

PE/IAE121, 213, 267, 320, 359, 421,505

Documents

TRIGONOMETRY - McDougal · PDF fileTrigonometry is designed to meet the needs of a trigonometry course covering one semester. The text introduces a unit-circle approach first and then