Chapter 6 knowing that the expression is the right side of the formula for

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Ch.6 AnalyticTrigonometry

6.1 VerifyingTrigonometricIdentities

1 UsetheFundamentalTrigonometricIdentitiestoVerifyIdentities

MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.

Completetheidentity.

1) secx–1

secx=?

A) sinxtanxB)1+cotxC)

–2tan2x D) secx cscx

2) cscx(sinx+cosx)=?

A) 1+cotx B) sinx tanxC)

–2tan2x D) secx cscx

3) sinx

cosx+cosx

sinx=?

A) secxcscxB)

–2tan2xC)1+cotx D) sinx tanx

4) (sinx+cosx)2

1+2sinxcosx=?

A) 1 B) 0 C) –sec2xD)1–sinx

5) 2tanx–(1+tanx)2=?

A) –sec2xB)1 C)0 D)1–sinx

6) tanx(cotx–cosx)=?

A) 1–sinxB)

–sec2x C) 1 D) 0

7) sin2x+sin2xcot2x=?

A) 1 B) sin2x+1 C) cot2x+1 D) cot2x–1

8) sin2x+tan2x+cos2x=?

A) sec2x B) tan2x C) cot3x D) sinx

9) cscxcotx

secx=?

A) cot2x B) 1 C) csc2x D) sec2x

10) sec4x+sec2xtan2x–2tan4x=?

A) 3sec4x–2B)4sec4x C) sec4x+2 D) tan2x–1

11) tanx·cotx=?

A) 1 B) –1 C) 0 D) sinx

12) (tanx+1)(tanx+1)–sec2x

tanx=?

A) 2 B) 1 C) 0 D) tan x

Page1

13) cosx–sinx

cosx+sinx–cosx

sinx=?

A) 2–secxcscxB)1–secx cscxC)2+secx cscx D) secx cscx

14) cos2x–sin2x

1–tan2x=?

A) cos2x B) sin2xC)1 D)

–1

15) sinx+cosx

sinx–cosx–sinx

cosx=?

A) secxcscxB)1–secx cscxC)2+secx cscxD)2–secx cscx

16) sin4x–cos4x=?

A) 1–2cos2xB)1+2cos2xC)1+2sin2xD)1–2sin2x

17) (cscx+1)(cscx–1)

cot2x=?

A) 1 B) 2 C) –1D)0

18) cot22x+cos22x+sin22x=?

A) csc22x B) sin22x C) cos22x D) 2

19) 1–sin2x

1+cosx=?

A) cosx B) 0 C) tanx D) cotx

20) 1–sinx

cosx=?

A) secx–tanx B) secx+tan xC)

–sec x–tan x D) sec x–tan x+1

21) csc2xsec x=?

A) sec x+csc xcotx B) sec x–csc xcotx C) csc xcotx–sec x D) secx+cscx

SHORTANSWER.Writethewordorphrasethatbestcompleteseachstatementoranswersthequestion.

Verifytheidentity.

22) cotθ·secθ=cscθ

23) tanθ·cscθ=secθ

24) csc2u–cosusecu=cot2u

25) (1+tan2u)(1–sin2u)=1

26) cscu–sinu=cosucotu

27) 1+sec2xsin2x=sec2x

28) cot 2x+csc 2x=2csc 2x–1

Page2

2 Tech:TrigonometricIdentities

MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.

Showthattheequationisnotanidentitybyfindingavalueofxforwhichbothsidesaredefinedbutnotequal.

1) cos(x–π)=cosx

A) 0 B) π

2C) –π

2D) 3π

2) sin(x+π)=sinx

A) π

2B) 0 C) 2πD) π

3) cosx–cosxsinx=cos3x

A) π

4B) π

2C) πD) 0

3 AdditionalConcepts

MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.

Usethegraphtocompletetheidentity.

1) (secx+tanx)(secx–tanx)

secx=?

-2–

2

-3

-2–

2

-3

A) cosx B) sinx C) cscx D) secx

2) sec2xcscx

sec2x+csc2x=?

-2–

2

-3

-2–

2

-3

A) sinx B) cosx C) cscx D) secx

Page3

3) cosxtanx–4tanx+3cosx–12

tanx+3=?

-2–

2

-3

-2–

2

-3

A) cosx–4 B) cosx+4 C) sinx–3 D) sinx+3cosx

4) 1+cosx

sinx+sinx

1+cosx=?

-2–

2

-3

-2–

2

-3

A) 2cscx B) 2secx C) 2sinx D) 2cosx

Rewritetheexpressionintermsofthegivenfunctionorfunctions.

5) (secx+cscx)(sinx+cosx)–2–cotx;tan x

A) tanxB)2+tan x C) 2tan xD)0

6) cos2x+cosx–1+sin2x;cosx

A) cosxB)

–cos xC)2+cos x D) 2cos x

7) cscx+tan2xcscx;cosxandsinx

A) 1

sinxcos2xB) sinx+cosx

sinxcosxC) 1

sinxcosxD) cosx–sinx

6.2 SumandDifferenceFormulas

1 UsetheFormulafortheCosineoftheDifferenceofTwoAngles

MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.

Findtheexactvalueoftheexpression.

1) cos(265°–25°)

A) –1

2B) –3

2C) –53

12 D) 3

Page4

2) cos5π

12 –π

A) 3

2B) 1

2C) 1

4D) 1

Identify

andβinthefollowingexpressionwhichistherightsideoftheformulaforcos(

–β).

3) cos(155°)cos(35°)+sin(155°)sin(35°)

A) α=155°

β=35° B) α=–155°

β =35° C) α= – 35°

β =155° D) α=35°

β =155°

4) cos5π

12 cosπ

4+sin5π

12 sinπ

A) α=5π

12 ,β=π

4B) α=5π

12 ,β=–π

4C) α=–π

4,β=5π

12 D) α=π

4,β=5π

Writetheexpressionasthecosineofanangle,knowingthattheexpressionistherightsideoftheformulaforcos(

–β)

withparticularvaluesforαandβ.

5) cos(170°)cos(50°)+sin(170°)sin(50°)

A) cos(120°) B) cos(210°) C) cos(190°) D) cos(220°)

6) cos5π

12 cosπ

4+sin5π

12 sinπ

A) cosπ

6B) cosπ

3C) cos2π

3D) cos5π

Findtheexactvalueoftheexpression.

7) cos(155°)cos(35°)+sin(155°)sin(35°)

A) –1

2B) –3

2C) –3 D) –2

Completetheidentity.

8) cos(α–β)

cosαcosβ =?

A) 1+tanαtanβ B) 1+cotαcotβ C) 1 +cotαtanβ D) tanαtanβ +cotβ

9) cosx–11π

6=?

A) 1

2(3cosx–sinx) B) 3

2(cosx–sinx)

C) –3

2(cosx+sinx) D) –3

2(cosx–sinx)

10) cos(2π–x)=?

A) cosxB)

–cos x C) sin xD)

–sin x

Usethegiveninformationtofindtheexactvalueoftheexpression.

11) sinα=4

5,αliesinquadrantII,andcosβ=2

5,βliesinquadrantI Findcos(α–β).

A) –6+421

25 B) 8+321

25 C) 8–321

25 D) 6–421

Page5

2 UseSumandDifferenceFormulasforCosinesandSines

MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.

Findtheexactvaluebyusingasumordifferenceidentity.

1) sin(215°–95°)

A) 3

2B) –1

2C) –43

12 D) –3

2) sin255°

A) –2(3+1)

4B) –2(3–1)

4C) 2(3+1)

4D) 2(3–1)

3) cos(45°+60°)

A) –2(3–1)

4B) –2(3+1)

4C) 2(3–1)

4D) 2(3+1)

4) cos285°

A) 2(3–1)

4B) –2(3–1)

4C) –2(3+1) D) –2(3–1)

Findtheexactvalueoftheexpression.

5) sin15°cos45°+cos15°sin45°

A) 3

2B) 1

2C) 1

4D) 3

6) sin245°cos5°–cos245°sin5°

A) –3

2B) –1

2C) –49

12 D) 3

7) sin220°cos100°–cos220°sin100°

A) 3

2B) –1

2C) –11

3D) –3

8) cos10°cos50°–sin10°sin50°

A) 1

2B) 3

2C) 1

4D) 3

9) cos7π

18 sin2π

9–cos2π

9sin7π

A) 1

2B) 3

2C) 1

4D) 1

Completetheidentity.

10) sinx+π

2=?

A) cosxB)

–cos x C) sin xD)

–sin x

11) cos(α+β)+cos(α–β)=?

A) 2cosαcosβ B) cosα cosβ C) 2sin αcosβ D) sin βcosα

Page6

12) sin(α+β)sin(α–β)=?

A) cos2β–cos2α B) sin2α–cos2β C) cos2β+cos2α D) sin2β–sin2α

13) sin(α+β)

cosαcosβ =?

A) tanα+tanβ B) tanβ +tanα C) cotα +cotβ D) –tanα +cotβ

14) sin(α+β)

sin(α–β)=?

A) tanα+tanβ

tanα–tanβ B) tanα–tanβ

tanα+tanβ C) tan (α+β)

tanα–tanβ D) tanα–tanβ

tan(α+β)

SHORTANSWER.Writethewordorphrasethatbestcompleteseachstatementoranswersthequestion.

Verifytheidentity.

15) cos x+π

2=–sinx

16) sin 3π

2–θ =–cosθ

17) cos 3π

2–θ =–sinθ

18) cos(α+β)

cosαsinβ =cotβ–tanα

19) sin(α+β)–sin(α–β)=2cosαsinβ

20) cos(α–β)–cos(α+β)=2sinαsinβ

21) sin(α–β)cos(α+β)=sinαcosα–sinβcosβ

MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.

Usethegiveninformationtofindtheexactvalueoftheexpression.

22) sinα=7

25 ,αliesinquadrantI,andcosβ=8

17 ,βliesinquadrantI Findcos(α+β).

A) 87

425 B) 297

425 C) –304

425 D) 416

425

23) sinα=20

29 ,αliesinquadrantII,andcosβ=5

13 ,βliesinquadrantI Findsin(α–β).

A) 352

377 B) 345

377 C) –135

377 D) –152

377

24) tanα=15

8,αliesinquadrantIII,andcosβ=–21

29 ,βliesinquadrantII Findsin(α+β).

A) 155

493 B) 468

493 C) –132

493 D) 475

493

Page7

25) sinα=4

5,αliesinquadrantII,andcosβ=2

5,βliesinquadrantI Findcos(α–β).

A) –6+421

25 B) 8+321

25 C) 8–321

25 D) 6–421

26) sinα=–3

5,αliesinquadrantIV,andcosβ=–21

5,βliesinquadrantIII Findsin(α–β).

A) 8+321

25 B) –8+321

25 C) –6+421

25 D) –6–421

27) cosα=–3

5,αliesinquadrantIII,andsinβ=21

5,βliesinquadrantII Findcos(α+β).

A) 6+421

25 B) –6–421

25 C) 8–321

25 D) –8+321

Describethegraphusinganotherequation.

28) y=sin(π–x)

2

-3

2

-3

A) sinx B) cosxC)

–sinxD)

–cosx

29) y=cosx+π

2–cosx–π

2

-3

2

-3

A) –2sinxB)

–2cosxC)

–sinxD)

–cosx

Page8

3 UseSumandDifferenceFormulasforTangents

MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.

Findtheexactvaluebyusingadifferenceidentity.

1) tan255°

A) 3+2B)

–3+2C)

–3–2D)3–2

2) tan285°

A) –2–3B) 2+3 C) 2–3

4D) 2+3

3) tan13π

A) 2–3 B) 2+3 C) 3–2D)

–2–3

Usetrigonometricidentitiestofindtheexactvalue.

4) tan25°+tan5°

1–tan25°tan5°

A) 3

3B) 3 C) 1

2D) 2

5) tan80°+tan70°

1–tan80°tan70°

A) –3

3B) –3 C) –1

2D) –2

6) tan175°–tan55°

1+tan175°tan55°

A) –3 B) –3

3C) –1

2D) –2

7) tan65°–tan(–55°)

1+tan65°tan(–55°)

A) –3 B) –3

3C) –1

2D) –2

Completetheidentity.

8) tan(π–θ)=?

A) –tanθ B) tanθ C) –cotθ D) cotθ

9) tan(π

4–x)=?

A) cosx–sinx

cosx+sinxB) cosx+sinx

cosx–sinxC) –cosx–sinx

cosx+sinxD) 1+cosx–sinx

cosx+sinx

SHORTANSWER.Writethewordorphrasethatbestcompleteseachstatementoranswersthequestion.

Verifytheidentity.

10) tan x–π

4=tanx–1

1+tanx

Page9

11) tan π

2+x=–cotx

MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.

Findtheexactvalueunderthegivenconditions.

12) sinα=5

13 ,0<α<π

2;cosβ=8

17 ,0<β<π

2Findtan(α+β).

A) 220

21 B) 171

221 C) 21

221 D) 220

221

13) tanα=8

15 ,π<α<3π

2;cosβ=–21

29 , π

2<β<π Findtan(α+β).

A) –132

475 B) 475

493 C) 31

95 D) –132

493

14) cosα=–4

5, π

2<α<π;sinβ=–21

5,π<β<3π

2Findtan(α+β).

A) –6+421

8+321 B) 8+321

–6+421 C) 6–421

8+321 D) –6+421

4 AdditionalConcepts

MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.

Rewritetheexpressionasasimplifiedexpressioncontainingoneterm.

1) cos5π

3+α cos5π

3–α –sin5π

3+α sin5π

3–α

A) –1

2B) 3

4C) 1

4D) 1

2) sin–5π

6–α cos–5π

6+α +cos–5π

6–α sin–5π

6+α

A) 3

2B) –1

2C) 1

2D) –3

Usethegraphtocompletetheidentity.

3) cos2xcos4x+sin2xsin4x=?

-2–

2

-3

-2–

2

-3

A) y=cos2x B) y=sin2x C) y=cos6x D) y=sin6x

Page10

4) sin6xcos3x–cos6xsin3x=?

-2–

2

-3

-2–

2

-3

A) y=sin3x B) y=cos3x C) y=cos9x D) y=sin9x

5) cos3xcos3

2x+sin3xsin3

2x=?

-2–

2

-3

-2–

2

-3

A) y=cos3

2xB)y=siny=cos3

2xC)y=cosy=cos9

2xD)y=siny=cos9

6) sin3xcos3

2x–cos3xsin3

2x=?

-2–

2

-3

-2–

2

-3

A) y=sin3

2xB)y=cos3

2xC)y=cos9

2xD)y=sin9

Page11

6.3 Double–Angle,Power–Reducing,andHal

–AngleFormulas

1 UseTheDouble–AngleFormulas

MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.

Usethefiguretofindtheexactvalueofthetrigonometricfunction.

1) Findcos2θ.

 

4

3

A) –7

25 B) 7

25 C) 24

25 D) –6

2) Findsin2θ.

 

3

4

A) 24

25 B) –7

25 C) 7

25 D) 8

3) Findtan2θ.





5

12

A) 120

119 B) 119

169 C) 120

169 D) 118

119

Usethegiveninformationtofindtheexactvalueoftheexpression.

4) sinθ=3

5,θliesinquadrantI Findcos2θ.

A) 7

25 B) –7

25 C) 24

25 D) 1

5) cosθ=21

29 ,θliesinquadrantIV Findsin2θ.

A) –840

841 B) –41

841 C) 41

841 D) 840

841

Page12

6) tanθ=20

21 ,θliesinquadrantIII Findsin2θ.

A) 840

841 B) –41

841 C) 41

841 D) –840

841

7) sinθ=7

25 ,θliesinquadrantII Findtan2θ.

A) –336

527 B) 336

527 C) 336

625 D) –529

527

Writetheexpressionasthesine,cosine,ortangentofadoubleangle.Thenfindtheexactvalueoftheexpression.

8) 2sin60°cos60°

A) 3

2B) –1

2C) –3

2D) 1

9) cos215°–sin215°

A) 3

2B) 1

2C) –1

2D) –3

10)

2tanπ

1–tan2π

A) 3 B) –3 C) 3

3D) –3

Completetheidentity.

11) 2cotx

1+cot2x=?

A) sin2x B) cos2x C) cot 2x D) tan 2x

12) (sinx–cosx)2=?

A) 1–sin2x B) sin2x C) 1–cos2x D) 1+cos2x

13) 1+cos2x

sin2x =?

A) cotx B) tanx C) cos2x D) sin2x

14) sin2x–cotx=?

A) –cotxcos2x B) cotxcos2x C) –tan xcos2x D) cot xcosx

15) 8sinxcos3x+8sin3xcosx=?

A) 4sin2x B) 4cos2x C) 4 sinxD)4cosx

SHORTANSWER.Writethewordorphrasethatbestcompleteseachstatementoranswersthequestion.

Verifytheidentity.

16) tan 2x(1+cos2x)=1–cos2x

17) sin4t=2sin2tcos2t

Page13

18) cos4θ=2cos2(2θ)–1

MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.

Solvetheproblem.

19) Ifaprojectileisfiredatanangleθandinitialvelocityv,thenthehorizontaldistancetraveledbytheprojectile

isgivenbyD=1

16 v2sinθcosθ.ExpressDasafunctionof2θ.

A) D=1

32 v2sin2θB) D=1

8v2sin2θC) D=1

16 v2sin2θD) D=1

16 v2cos2θ

2 UsethePower–ReducingFormulas

MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.

Rewritetheexpressionasanequivalentexpressionthatdoesnotcontainpowersoftrigonometricfunctionsgreaterthan

1) 8cos2x

A) 4+4cos2x B) 4–4cos2x C) 1+cos2x D) 16cosx

2) sin4x

A) 3

8–1

2cos2x+1

8cos4x B) 3

2–3

2cos2x

C) 3

2–2cos2x+1

2cos4x D) 3

8+5

8cos2x

3) cos4x

A) 3+4cos2x+cos4x

8B) 3+2cos2x+cos4x

C) 3–4cos2x+cos4x

8D) 2–4cos2x+cos4x

4) sin3x

A) 3

4sinx–1

4sin3x B) 3

4sinx+1

2sin2x–1

4sin3x

C) 3

4sinx+1

4sin2x D) 3

4sinx

5) cos3x

A) 3

4cosx+1

4cos3x B) 3

4cosx–1

4cos3x–cos2x

C) 3

4cosx–1

4cos3x D) 3

4cosx+1

4cos3x+cos2x

6) 3sin2xcos2x

A) 3

8–3

8cos4x B) 3

4–3

4cos4x C) 3

8–3

8cos2x D) 3

4sin2x

Page14

3 UseTheHalf–AngleFormulas

MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.

Useahalf–angleformulatofindtheexactvalueoftheexpression.

1) sin105°

A) 1

22+3 B) 1

22–3 C) –1

22+3 D) –1

22–3

2) sin5π

A) 1

22+3 B) 1

22–3 C) –1

22+3 D) –1

22–3

3) cos5π

A) 1

22–3 B) 1

22+3 C) –1

22+3 D) –1

22–3

4) cos3π

A) 1

22–2 B) 1

22+2 C) –1

22+2 D) –1

22–2

5) sin3π

A) 1

22+2 B) 1

22–2 C) –1

22+2 D) –1

22–2

6) tan112.5°

A) –2–1B)2+1C)

–2+1D)2–1

Usethegiveninformationtofindtheexactvalueofthetrigonometricfunction.

7) sinθ=1

4,θliesinquadrantI Findsinθ

A) 8–215

4B) 10

4C) 6

4D) 8+215

8) sinθ=1

4,tanθ>0 Findcosθ

A) 8+215

4B) 10

4C) 6

4D) 8–215

9) cosθ=1

4,cscθ>0 Findsinθ

A) 6

4B) 8+215

4C) 10

4D) 8–215

Page15

10) secθ=4,θliesinquadrantI Findcosθ

A) 10

4B) 6

4C) 8–215

4D) 8+215

11) cosθ=–3

5,θliesinquadrantIII Findcosθ

A) –5

5B) 5

5C) –30

10 D) 30

12) cosθ=–3

5,sinθ>0 Findcosθ

A) 5

5B) –5

5C) –30

10 D) 30

13) sinθ=–3

5,θliesinquadrantIV Findsinθ

A) 10

10 B) 5

5C) –5

5D) –30

14) secθ=–5

3,θliesinquadrantII Findsinθ

A) 25

5B) –5

5C) –2

5D) 5

15) sinθ=–15

17 ,θliesinquadrantIV Findcosθ

A) –534

34 B) 534

34 C) 4

17 D) –334

16) cscθ=–6

5,tanθ>0 Findcosθ

A) –18–311

6B) 18+311

6C) 11

12 D) –6+11

17) cosθ=4

5,θliesinquadrantIV Findcosθ

A) –310

10 B) 2

5C) 310

10 D) –2

Completetheidentity.

18) sin2θ

2=?

A) cscθ–cotθ

2cscθ B) cscθ +cotθ

2cscθ C) –csc θ–cotθ

2cscθ D) –csc θ+cotθ

2cscθ

Page16

19) cotθ

2=?

A) sinθ

1–cosθ B) sin θ

1+cosθ C) sin θ

sinθ–cosθ D) sin θ

sinθ+cosθ

20) tanθ

2+cotθ

2=?

A) 2cscθ B) –2csc θC) –2 cot θD) 2cot θ

Solvetheproblem.

21) Anairplaneflyingfasterthanthespeedofsoundcreatessoundwavesthatformacone.Ifαisthevertexangle

oftheconeandmistheMachnumberforthespeedoftheplane,thensinα

2=1

m(m>1).FindtheMach

numbertothenearesttenthifα=90°.

A) 1.4 B) 1.0 C) 1.7 D) 2.4

4 Tech:DoubleandHalf–AngleFormulas

MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.

Useagraphina[–2π,2π,π

2]by[–3,3,1]viewingrectangletocompletetheidentity.

1) 1–2cos2x

2sinx–1=?

A) 2sinx+1B)2cosx+1 C) sinx+1 D) cosx–1

2tanx

1+tan2x

=?

A) sinx B) cosx C) sinx+1D)2cosx–1

6.4 Product–to–SumandSum–to–ProductFormulas

1 UsetheProduct–toSumFormulas

MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.

Fillintheblankusingthewordproduct,sum,quotient,ordifference.

1) Theformulasinαcosβ=1

2[sin(α+β)+sin(α–β)]canbeusedtochangea ofasineandacosine

intotheoftwosineexpressions.

A) product,sum B) product,difference

C) quotient,sum D) quotient,difference

Expresstheproductasasumordifference.

2) sin7xcos4x

A) 1

2(sin11x+sin3x) B) sin(cos28x2)

C) 1

2(cos11x–cos3x) D) 1

2(sin11x+cos3x)

Page17

3) sin6xsin3x

A) 1

2(cos3x–cos9x) B) sin218x2C) 1

2(cos9x–cos3x) D) 1

2(sin9x+cos3x)

4) cos6xcos5x

A) 1

2(cosx+cos11x) B) 1

2(cos11x–sinx) C) 1

2(cos11x–cosx) D) cos230x2

5) sin5xsin9x

A) 1

2(cos4x–cos14x) B) 1

2(cos14x–sin4x)

C) 1

2(–cos4x–cos14x) D) sin245x2

6) sin3xcos4x

A) 1

2(sin7x–sinx) B) 1

2(cos7x+sinx) C) 1

2(cos7x–cosx) D) sin(cos12x2)

7) cos3xcos7x

A) 1

2(cos4x+cos10x) B) 1

2(cos10x–sin4x)

C) 1

2(cos10x–cos4x) D) cos213x2

8) cos5x

2cosx

A) 1

2(cos2x+cos3x) B) 1

4(cos6x–sin4x) C) 1

4cos25x D) 1

2(cos3x–sin2x)

9) sinx

2cos11x

A) 1

2(sin6x–sin5x) B) 1

4(cos12x–sin10x)

C) 1

4sin(cos11x) D) 1

2(cos6x+sin5x)

Completetheidentity.

10) sin3xsin5xcos3xcos5x=?

A) cos22x–cos28x

4B) sin230x

4C) cos28x+cos22x

4D) cos230x

11) sinx(sin5x+sin7x)=?

A) cosx(cos5x–cos7x) B) cosx (cos5x +cos7x)

C) 1

2cosx(cos5x–cos7x) D) 1

2cosx(cos5x+cos7x)

Page18

2 UsetheSum–to–ProductFormulas

MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.

Fillintheblankusingthewordproduct,sum,quotient,ordifference.

1) Theformulacosα–cosβ=–2sinα+β

2sinα–β

2canbeusedtochangea oftwocosinesintothe

oftwosineexpressions.

A) difference

product B) sum

product

C) difference

quotient D) sum

quotient

Expressthesumordifferenceasaproduct.

2) sin6x+sin2x

A) 2sin4xcos2x B) 2cos4x sin2x C) 2sin4x sin2x D) 2sin8x

3) cos10x–cos4x

A) –2sin7xsin3x B) 2cos7x cos3x C) –2cos7x sin3x D) 2cos3x

4) cos6x+cos4x

A) 2cos5xcosxB)2sin5x sinxC)2cos5x sinxD)2cos5x

5) cos2x–cos4x

A) 2sin3xsinxB)

–2sin3x sinxC)

–2cos3x sinx D) cos(–2x)

6) sin6x–sin2x

A) 2sin2xcos4x B) 2cos2x cos4x C) 2sin4x cos2x D) 2sin2x

7) sin4x–sin6x

A) –2sinxcos5x B) 2cos4x cos5x C) 2sin5x cosxD)

–2sinx

8) cos5x

2+cos3x

A) 2cos2xcosx

2B) 2sin2x sinxC)2sin2xsinx

2D) 2cos2x

9) sin7x

2+sin3x

A) 2sin5x

2cosxB)2sin5x

2sinxC)2cos5x sinxD)2sin5x

10) sin75∘+sin15∘

A) 6

2B) 6

4C) 3

2D) –6

Completetheidentity.

11) sin5x+sin9x

cos5x+cos9x =?

A) tan7x B) tan5x +tan9x C) 2tan7x tan2x D) tan7x cot2x

12) cos2x–cos8x

cos2x+cos8x =?

A) tan3xtan5x B) –tan5x C) cot5x D) 0

Page19

13) sin5x+sin7x

cos2x+cos4x =?

A) sin6x

cos3x B) tan5

2x+tan7

4xC)

sin12x

cos6x D) cos12x

sin6x

14) sinx+siny

cosx+cosy=?

A) tanx+y

2B) tanx+tany C) tanx–y

2D) cotx–y

SHORTANSWER.Writethewordorphrasethatbestcompleteseachstatementoranswersthequestion.

Verifytheidentity.

15) cosx+cosy

sinx–siny=cotx–y

16) sinα–sinβ

sinα+sinβ =tanα–β

2cotα+β

MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.

Solvetheproblem.

17) Thesoundproducedbytouchingeachbuttononatouch–tonephoneisdescribedby

y=sin2πlt+sin2πht

wherelandharethelowandhighfrequencies(cyclespersecond)inthefigureshown.

Describethesoundemittedbytouchingthe8keyasaproductofsinesandcosines.

A) y=2sin(2188πt)cos(484πt) B) y=2sin(2329πt) cos(625πt)

C) y=2sin(484πt)cos(2188πt) D) y=2sin(625πt)cos(2329πt)

3 Tech:Product–to–SumandSum–to–ProductFormulas

MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.

Showthattheequationisnotanidentitybyfindingavalueofxforwhichbothsidesaredefinedbutnotequal.

1) sin4x+sin6x=sin9x

A) π

2B) 0 C) πD) 2π

2) cos3x+cos8x=2cos5xcosx

A) π

4B) 0 C) 2πD) –2π

Page20

Solvetheproblem.

3) Thesoundproducedbytouchingeachbuttononatouch–tonephoneisdescribedby

y=sin2πlt+sin2πht

wherelandharethelowandhighfrequencies(cyclespersecond)inthefigureshown.

Useacalculatortofindthegraphofthesoundemittedbytouchingthe#keyina[0,0.01,0.001]by[–2,2,1]

viewingrectangle.

0.005 0.01

-2

0.005 0.01

-2

0.005 0.01

-2

0.005 0.01

-2

0.005 0.01

-2

0.005 0.01

-2

0.005 0.01

-2

0.005 0.01

-2

Page21

4) Graphina[–π,π,π

2]by[–3,3,1]viewingrectangle

y=2sinx

1–sin2x

2+sin3x

3–sin4x

4+sin5x

5–sin6x

6+sin7x

––



2

-1

-2

-3

––



2

-1

-2

-3

––



2

-1

-2

-3

––



2

-1

-2

-3

––



2

-1

-2

-3

––



2

-1

-2

-3

––



2

-1

-2

-3

––



2

-1

-2

-3

4 AdditionalConcepts

MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.

Usethegraphtocompletetheidentity.

1) cos8x–cos2x

sin8x+sin2x =?

2

-3

2

-3

A) –tan3x B) cot2x +cot8x C) 2tan5x tan3x D) tan5x

Page22

2) sin2x+sin6x

cos6x–cos2x =?

2

-3

2

-3

A) –cot2x B) tan2x +tan6x C) cot2x D) tan4x

6.5 TrigonometricEquations

1 FindAllSolutionsofaTrigonometricEquation

MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.

Usesubstitutiontodeterminewhetherthegivenx–valueisasolutionoftheequation.

1) sinx=–3

2,x=4π

A) Yes B) No

2) tanx=3

3,x=7π

A) Yes B) No

3) sinx=–23

3,x=–2π

A) Yes B) No

4) cos2x=–2,x=3π

A) Yes B) No

5) cosx+1=sinx,x=–3π

A) Yes B) No

Findallsolutionsoftheequation.

6) cosx=0

A) π

2+nπB) x=π

2+2nπC) 2nπD) nπ

Page23

7) 2cosx–3=0

A) x=π

6+2nπorx=11π

6+2nπB) x=π

6+nπorx=11π

6+nπ

C) x=5π

6+2nπorx=7π

6+2nπD) x=5π

6+nπorx=7π

6+nπ

8) 8cosx–63

=6cosx–53

A) x=π

6+2nπorx=11π

6+2nπB) x=π

6+nπorx=11π

6+nπ

C) x=5π

6+2nπorx=7π

6+2nπD) x=5π

6+nπorx=7π

6+nπ

9) tanxsecx=–2tanx

A) x=2π

3+2nπorx=4π

3+2nπorx=nπB) x=2π

3+nπorx=4π

3+nπorx=nπ

C) x=π

3+2nπorx=5π

3+2nπorx=nπD) x=π

3+nπorx=5π

3+nπorx=nπ

2 SolveEquationsWithMultipleAngles

MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.

Solvetheequationontheinterval[0,2π).

1) sin4x=3

A) π

12 ,π

6,2π

3,7π

12 ,7π

6,13π

12 ,5π

3,19π

12 B) π

4,5π

C) 0,π

4,π D) 0

2) cos2x=2

A) π

8,7π

8,9π

8,15π

8B) 0,2π

3,π,4π

3C) π

4,3π

4,5π

4,7π

4D) nosolution

3) cos2x=3

A) π

12 ,11π

12 ,13π

12 ,23π

12 B) π

6,11π

C) π

2D) 3π

3 SolveTrigonometricEquationsQuadraticinForm

MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.

Solvetheequationontheinterval[0,2π).

1) cos2x+2cosx+1=0

A) πB) 2πC) π

2,3π

2D) π

4,7π

Page24

2) 2sin2x=sinx

A) 0,π,π

6,5π

6B) π

2,3π

2,π

3,2π

3C) π

6,5π

6D) π

3,2π

3) cosx=sinx

A) π

4,5π

4B) π

4,7π

4C) 3π

4,5π

4D) 3π

4,7π

4) sec2x–2=tan2x

A) nosolution B) π

6C) π

4D) π

5) tanx+secx=1

A) 0 B) π

4C) 5π

4D) nosolution

6) csc5x–4cscx=0

A) π

4,3π

4,5π

4,7π

4B) π

4,3π

4,π

6,5π

6C) π

4,5π

4,π

3,5π

3D) π

4,3π

4,π

3,5π

7) sin2x–cos2x=0

A) π

4,3π

4,5π

4,7π

4B) π

4,π

6C) π

4,π

3D) π

8) sin2x+sinx=0

A) 0,π,3π

2B) 0,π,4π

3,5π

3C) 0,π,π

3,5π

3D) 0,π,π

3,2π

4 UseFactoringtoSeparateDifferentFunctionsinTrigonometricEquations

MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.

Solvetheequationontheinterval[0,2π).

1) –tan2xsinx=–tan2x

A) 0,π B) π,2πC) π,3π

2D) 0,3π

2) (tanx–1)(cosx+1)=0

A) π

4,π,5π

4B) 0,π

4,5π

4C) 0,3π

4,7π

4D) 3π

4,π,7π

3) cosx+2cosxsinx=0

A) π

2,7π

6,3π

2,11π

6B) 0,π

2,7π

6,3π

2C) 7π

6,11π

6D) 7π

6,11π

6,2π

Page25

5 UseIdentitiestoSolveTrigonometricEquations

MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.

Solvetheequationontheinterval[0,2π).

1) tan2x–tanx=0

A) 0,π B) π

4,5π

C) π

12 ,π

6,2π

3,7π

12 ,7π

6,13π

12 ,5π

3D) 0

2) secx

2=cosx

A) 0 B) π

4,5π

C) π

12 ,π

6,2π

3,7π

12 ,7π

6,13π

12 ,5π

3D) 0,π

4,π,5π

3) sin3x=0

A) 0,π

3,2π

3,π,4π

3,5π

3,2πB) π

4,3π

4,5π

4,7π

C) π

8,9π

8D) nosolution

4) sin22x=1

A) π

4,3π

4,5π

4,7π

4B) 0,2π

3,π,4π

3C) π

8,9π

8D) nosolution

5) cos2x=2–cos2x

A) π

8,7π

8,9π

8,15π

8B) 0,2π

3,π,4π

3C) π

4,3π

4,5π

4,7π

4D) nosolution

6) sin2x+sinx=0

A) 0,2π

3,π,4π

3B) π

8,9π

8C) π

4,3π

4,5π

4,7π

4D) nosolution

7) 2cos2x+sinx–2=0

A) 0,π,π

6,5π

6B) π

2,3π

2,π

3,2π

3C) π

6,5π

6D) π

3,2π

8) sin x+5π

4+sin x–5π

4=1

A) 5π

4,7π

4B) 3π

2,3πC) 3π

4,5π

6D) 5π

4,5π

9) cos x+π

4–cos x–π

4=1

A) 5π

4,7π

4B) π

2,π

6C) π

6,π

3D) π

4,π

Page26

10) cos x+π

3+cos x–π

3=1

A) 0 B) π

2,π

6C) π

6,π

4D) π

2,π

11) sin x+7π

6–sin x–7π

6=1

A) πB) π

2,π

4C) π

4,π

3D) π

2,π

6 UseaCalculatortoSolveTrigonometricEquations

MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.

Useacalculatortosolvetheequationontheinterval[0,2π).Roundtheanswertotwodecimalplaces.

1) sinx=0.29

A) 0.29

2.85 B) 0.29

5.99 C) 0.29

3.43 D) 0.29

1.86

2) cosx=0.81

A) 0.63

5.65 B) 0.63

2.51 C) 0.63

3.77 D) 0.63

2.20

3) tanx=2.4

A) 1.18

4.32 B) 1.18

5.10 C) 1.18

1.96 D) 1.18

2.75

4) sinx=–0.25

A) 3.39

6.03 B) 0.25

6.03 C) 0.25

3.39 D) 0.25

1.82

5) cosx=–0.72

A) 2.37

3.91 B) 2.37

5.52 C) 0.77

3.91 D) 0.77

2.37

Useacalculatortosolvetheequationontheinterval[0,2π).Roundtothenearesthundredthofaradian.

6) sin3x=–sinx

A) 0,1.57,3.14,4.71 B) 1.57,4.71

C) 0,3.14 D) 0,0.79,2.36,3.14,3.93,5.50

7) sin2x+sinx=0

A) 0,2.09,3.14,4.19 B) 0,1.05,3.14,5.24 C) 1.05,3.14,5.24 D) 0,2.09,4.19

7 SolveApps:TrigonometricEquations

MULTIPLECHOICE.Choosetheonealternativethatbestcompletesthestatementoranswersthequestion.

Solvetheproblem.

1) TheoutputvoltageforanACgeneratorisapproximatedbyv=156cos(120πt–π

3).Findthesmallestpositive

valueoftforwhichtheoutputis146volts.

A) 0.0037second B) 0.0027 second C) 0.0074 second D) 0.0084 second

2) Acoilofwirerotatinginamagneticfieldinducesavoltagegivenbye=20sin(πt

4–π

2),wheretistimein

seconds.Findthesmallestpositivetimetoproduceavoltageof10 3.

A) 3.33seconds B) 3seconds C) 3.33πseconds D) 3πseconds

Page27

3) AgeneratorproducesanalternatingcurrentaccordingtotheequationI=56 sin43πt,wheretistimein

secondsandIisthecurrentinamperes.WhatisthesmallesttimetsuchthatI=28?

A) 1

258 second B) 1

129 second C) 1

172 second D) 1

86 second

4) Theweeklysalesinthousandsofitemsofaproducthasaseasonalsalesrecordapproximatedby

n=53.6+19.3sinπt

24 (tistimeinweekswitht=1referringtothefirstweekintheyear).Duringwhichweek(s)

willthesalesequal63,250items?

A) week4,week20,andweek52 B) week30andweek47

C) week4andweek47 D) week21andweek30

5) Therangerofaprojectileisgivenbyr=1

32 v2sin2θ,wherevistheinitialvelocityandθistheangleof

elevation.Ifristobe10,000ftandv=3000ft/sec,whatmusttheangleofelevationbe?Giveyouranswerin

degreestothenearesthundredth.

A) 1.02° B) 88.98° C) 2.04° D) 1.43°

Page28

Ch.6 AnalyticTrigonometry

AnswerKey

6.1 VerifyingTrigonometricIdentities

1 UsetheFundamentalTrigonometricIdentitiestoVerifyIdentities

2 Tech:TrigonometricIdentities

3 AdditionalConcepts

Page29

6.2 SumandDifferenceFormulas

1 UsetheFormulafortheCosineoftheDifferenceofTwoAngles

2 UseSumandDifferenceFormulasforCosinesandSines

3 UseSumandDifferenceFormulasforTangents

4 AdditionalConcepts

6.3 Double–Angle,Power–Reducing,andHal

–AngleFormulas

1 UseTheDouble–AngleFormulas

2 UsethePower–ReducingFormulas

3 UseTheHalf–AngleFormulas

Page31

4 Tech:DoubleandHalf–AngleFormulas

6.4 Product–to–SumandSum–to–ProductFormulas

1 UsetheProduct–toSumFormulas

2 UsetheSum–to–ProductFormulas

Page32

3 Tech:Product–to–SumandSum–to–ProductFormulas

4 AdditionalConcepts

6.5 TrigonometricEquations

1 FindAllSolutionsofaTrigonometricEquation

9) A

2 SolveEquationsWithMultipleAngles

3 SolveTrigonometricEquationsQuadraticinForm

4 UseFactoringtoSeparateDifferentFunctionsinTrigonometricEquations

5 UseIdentitiestoSolveTrigonometricEquations

Page33

6 UseaCalculatortoSolveTrigonometricEquations

7 SolveApps:TrigonometricEquations

Page34