C H A P T E R 4 Trigonometric Functions - Fraser HS Math

C H A P T E R 4

Trigonometric Functions

Section 4.1 Radian and Degree Measure . . . . . . . . . . . . . . . . 272

Section 4.2 Trigonometric Functions: The Unit Circle . . . . . . . . 281

Section 4.3 Right Triangle Trigonometry . . . . . . . . . . . . . . . . 289

Section 4.4 Trigonometric Functions of Any Angle . . . . . . . . . . 300

Section 4.5 Graphs of Sine and Cosine Functions . . . . . . . . . . . 317

Section 4.6 Graphs of Other Trigonometric Functions . . . . . . . . . 329

Section 4.7 Inverse Trigonometric Functions . . . . . . . . . . . . . . 339

Section 4.8 Applications and Models . . . . . . . . . . . . . . . . . . 350

Review Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 360

Practice Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 377

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C H A P T E R 4

Trigonometric Functions

Section 4.1 Radian and Degree Measure

272

You should know the following basic facts about angles, their measurement, and their applications.

n Types of Angles:

(a) Acute: Measure between and

(b) Right: Measure

(c) Obtuse: Measure between and

(d) Straight: Measure

n and are complementary if They are supplementary if

n Two angles in standard position that have the same terminal side are called coterminal angles.

n To convert degrees to radians, use radians.

n To convert radians to degrees, use 1 radian

n one minute of

n one second of

n The length of a circular arc is where is measured in radians.

n Speed

n Angular speed 5 uyt 5 syrt

5 distanceytime

us 5 ru

185 1y60 of 19 5 1y360010 5

185 1y6019 5

5 s180ypd8.18 5 py180

a 1 b 5 1808.a 1 b 5 908.ba

1808.

1808.908

908.

908.08

1. The angle shown is

approximately 2 radians.

Vocabulary Check

1. Trigonometry 2. angle 3. standard position 4. coterminal

5. radian 6. complementary 7. supplementary 8. degree

9. linear 10. angular

2. The angle shown is

approximately radians.24

3. (a) Since lies in Quadrant IV.

(b) Since lies in

Quadrant II.

5p

2<

11p

4< 3p,

11p

4

3p

2<

7p

4< 2p,

7p

44. (a) Since lies in

Quadrant IV.

(b) Since lies in

Quadrant II.

23p

2< 2

13p

9< 2p,

13p

9

2p

2< 2

5p

12< 0,

5p

2

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Section 4.1 Radian and Degree Measure 273

9. (a)

(b)

x

y

π

32

2p

3

x

116π

y11p

6

7. (a)

(b)

x

y

π

34

4p

3

43π

x

y13p

4

5. (a) Since lies in Quadrant IV.

(b) Since lies in

Quadrant III.

2p < 22 < 2p

2; 22

2p

2< 21 < 0; 21 6. (a) Since 3.5 lies in Quadrant III.

(b) Since 2.25 lies in Quadrant II.p

2< 2.25 < p,

p < 3.5 <

3p

2,

8. (a)

(b)

x

y

52π

−

25p

2

x

74

−π

y

27p

4

10. (a) 4

(b)

x

y

−3

23

x

4

y

11. (a) Coterminal angles for

p

62 2p 5 2

11p

6

p

61 2p 5

13p

6

p

6: (b) Coterminal angles for

2p

32 2p 5 2

4p

3

2p

31 2p 5

8p

3

2p

3:

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274 Chapter 4 Trigonometric Functions


(b) Coterminal angles for

5p

42 2p 5 2

3p

4

5p

41 2p 5

13p

4

5p

4:

7p

62 2p 5 2

5p

6

7p

61 2p 5

19p

6

7p

6: 13. (a) Coterminal angles for


22p

152 2p 5 2

32p

15

22p

151 2p 5

28p

15

22p

15:

29p

41 4p 5

7p

4

29p

41 2p 5 2

p

4

29p



8p

452 2p 5 2

82p

45

8p

451 2p 5

98p

45

8p

45:

7p

82 2p 5 2

9p

8

7p

81 2p 5

23p

8

7p

8:

15. Complement:

Supplement: p 2p

35

2p

3

p

22

p

35

p

6

17. Complement:

Supplement: p 2p

65

5p

6

p

22

p

65

p

3

16. Complement: Not possible; is greater than

(a)Supplement: p 23p

45

p

4

p

2.

3p

4

18. Complement: Not possible; is greater than

Supplement: p 22p

35

p

3

p

2.

2p

3

19. Complement:

Supplement: p 2 1 < 2.14

p

22 1 < 0.57 20. Complement: None

Supplement: p 2 2 < 1.14

12 >

p

22

21.

The angle shown is approximately 2108.

22.

The angle shown is approximately 2458.


Quadrant II.

(b) Since lies in

Quadrant IV.

28282708 < 2828 < 3608,

1508908 < 1508 < 1808,

25. (a) Since

lies in Quadrant III.

(b) Since

lies in Quadrant I.23368 309

23608 < 23368 309 < 22708,

21328 509

21808 < 21328 509 < 2908,


Quadrant I.

(b) Since lies in Quadrant I.8.5808 < 8.58 < 908,

87.9808 < 87.98 < 908,

26. (a) Since

lies in Quadrant II.

(b) Since lies

in Quadrant IV.

212.3582908 < 212.358 < 208,

2245.25822708 < 2245.258 < 21808,

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27. (a)

308x

y308 (b)

150°

x

y1508

28. (a)

(b)

x

−120°

y

21208

x

−270°

y

22708 29. (a)

(b)

780°

x

y

7808

x

405°

y

4058 30. (a)

(b)

x

y

−6008

26008

− °450

x

y

24508



2368 2 3608 5 23968

2368 1 3608 5 3248

2368:

528 2 3608 5 23088

528 1 3608 5 4128



2308 2 3608 5 21308

2308 1 3608 5 5908

2308:

3008 2 3608 5 2608

3008 1 3608 5 6608

3008:32. (a) Coterminal angles for


23908 1 3608 5 2308

23908 1 7208 5 3308

23908:

1148 2 3608 5 22468

1148 1 3608 5 4748

1148:



27408 1 7208 5 2208

27408 1 10808 5 3408

27408:

24458 1 3608 5 2858

24458 1 7208 5 2758

24458: 35. Complement:

Supplement: 1808 2 248 5 1568

908 2 248 5 668 36. Complement: Not possible

Supplement: 1808 2 1298 5 518

37. Complement:

Supplement: 1808 2 878 5 938

908 2 878 5 38 38. Complement: Not possible

Supplement: 1808 2 1678 5 138

39. (a)

(b) 1508 5 15081 p

18082 55p

6

308 5 3081 p

18082 5p

6

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44. (a)

(b) 3p 5 3p11808

p 2 5 5408

24p 5 24p11808

p 2 5 27208

46. (a)

(b)28p

155

28p

15 11808

p 2 5 3368

215p

65 2

15p

6 11808

p 2 5 2450845. (a)

(b) 213p

605 2

13p

60 11808

p 2 5 2398

7p

35

7p

3 11808

p 2 5 4208

47. 1158 5 1151p

18082 < 2.007 radians 48. radians83.78 5 83.781 p

18082 < 1.461

50. radians246.528 5 246.5281 p

18082 < 20.81249. 2216.358 5 2216.351p

18082 < 23.776 radians

51. 20.788 5 20.781 p

18082 < 20.014 radians

53.p

75

p

711808

p 2 < 25.7148 55. 6.5p 5 6.5p11808

p 2 5 11708

61. 858 189 300 5 858 1 s1860d8 1 s 30

3600d8 < 85.3088

63. 21258 360 5 21258 2 s 363600d8 5 2125.018

65. 280.68 5 2808 1 0.6s60d9 5 2808 369

67. 2345.128 5 23458 79 120

57. 22 5 2211808

p 2 < 2114.5928

59. 648 459 5 648 1 145

6028

5 64.758

52. radians3958 5 39581 p

18082 < 6.894

54.8p

135

8p

1311808

p 2 < 110.7698

56. 24.2p 5 24.2p11808

p 2 5 27568

58. 20.48 5 20.4811808

p 2 < 227.5028

60. 21248 309 5 2124.58

62. 24088 169 250 < 2408.2748

64. 3308 250 < 330.0078

66. 2115.88 5 21158 489 68. 310.758 5 3108 459

43. (a)

(b) 27p

65 2

7p

6 11808

p 2 5 22108

3p

25

3p

2 11808

p 2 5 2708

40. (a)

(b) 1208 5 12081 p

18082 52p

3

3158 5 31581 p

18082 57p

4

42. (a) (b) 1448 5 14481 p

18082 54p

522708 5 227081 p

18082 5 23p

2

41. (a)

(b) 22408 5 224081 p

18082 5 24p

3

2208 5 22081 p

18082 5 2p

9

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69.

< 220.348 5 2208 209 240

20.355 5 20.35511808

p 2 70.

< 458 39 470

5 458 1 39 1 0.786s600 d

5 458 1 s0.0631ds609d

< 45.06318

0.7865 5 0.786511808

p 2

72.

radians u 53112 5 2

712

31 5 12u

s 5 ru71.

radiansu 565

6 5 5u

s 5 ru 73.

radians3u 5327 5 4

47

32 5 7u

3s 5 ru

74.

Because the angle represented is clockwise, this angle is radians.245

u 56075 5

45 radians

60 5 75u

s 5 ru

78. radianu 5s

r5

10

225

5

11

75. The angles in radians are:

908 5p

2

608 5p

3

458 5p

4

308 5p

6

08 5 0

3308 511p

6

2708 53p

2

2108 57p

6

1808 5 p

1358 53p

4

76. The angles in degrees are:

5p

65 1508

2p

35 1208

p

35 608

p

45 458

p

65 308

p 5 1808

7p

45 3158

5p

35 3008

4p

35 2408

5p

45 2258

77.

radians u 58

15

8 5 15u

s 5 ru 79.

radians u 570

29< 2.414

35 5 14.5u

s 5 ru

80. kilometers, kilometers

u 5s

r5

160

805 2 radians

s 5 160r 5 80

82. feet,

s 5 ru 5 91p

32 5 3p feet

u 5 608 5p

3r 5 9

81. in radians

inches s 5 14s180d1 p

1802 5 14p < 43.982

s 5 ru, u

83. in radians

meters meters< 56.55 s 5 2712p

3 2 5 18p

s 5 ru, u

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90.

s 5 ru 5 4000s1.0105d < 4042.0 miles

u 5 318 469 1 268 89 5 578 549 < 1.0105 rad

r 5 4000 miles

89.

s 5 ru 5 4000s0.28537d < 1141.48 miles

5 168 219 10 < 0.28537 radian

u 5 428 79 330 2 258 469 32088. r 5s

u5

8

3308spy1808d5

48

11p inches < 1.39 inches

91.

< 48 29 33.020

u 5s

r5

450

6378< 0.07056 radian < 4.048

93. < 23.878u 5s

r5

2.5

65

25

605

5

12 radian

95. (a) single axel:

(b) double axel:

(c) triple axel: 5 7p radians 312 revolutions 5 12608

5 4p 1 p 5 5p radians

212 revolutions 5 7208 1 1808 5 9008

5 2p 1 p 5 3p radians

112 revolutions 5 3608 1 1808 5 5408

96. Linear speed 5s

t5

ru

t5

s6400 1 1250d2p

110< 436.967 kmymin

92.

u 5s

r5

400

6378< 0.0627 rad < 3.598

r 5 6378, s 5 400

94. u 5s

r5

24

55 4.8 rad < 275.028

97. (a)

Angular speed

(b) Radius of saw blade

Radius in feet

Speed

5 0.3125s80pd 5 78.54 ftysec

5s

t5

ru

t5 r

u

t5 rsangular speedd

53.75

125 0.3125 ft

57.5

25 3.75 in.

5 s2pds40d 5 80p radysec

Revolutions

Second5

2400

605 40 revysec 98. (a)

Angular speed

(b) Radius of saw blade

Radius in feet

Speed

5 0.3021s160pd < 151.84 ftysec

5s

t5

ru

t5 r

u

t5 rsangular speedd

53.625

12 < 0.3021 ft

57.25

25 3.625 in.

5 s2pds80d 5 160p radysec

Revolutions

Second5

4800

605 80 revysec

86. r 5s

u5

3

4py35

9

4p meters < 0.72 meter 87. r 5

s

u5

82

1358spy1808d5

328

3p miles < 34.80 miles

84. centimeters,

centimeters cm< 28.27s 5 ru 5 1213p

4 2 5 9p

u 53p

4r 5 12 85. r 5

s

u5

36

py25

72

p feet < 22.92 feet

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99. (a)

Angular speed

Radius of wheel

Speed

(b) Let spin balance machine rate.

70 5 rsangular speedd 5 r12p ?x

s1y60d2 55

25,344120px ⇒ x < 941.18 revymin

x 5

55

25,344? 57,600p 5

125p

11< 35.70 milesyhr

5s

t5

ru

t5 r

u

t5 rsangular speedd

525y2

s12 in.yftds5280 ftymid5

5

25,344 miles

5 2p s28,800d 5 57,600p radyhr

Revolutions

Hour5

480

s1y60d5 28,800 revyhr

100. (a)

(b) Speed

For the outermost track, 6000p cmymin

6s400pd ≤ linear speed ≤ 6s1000pd

5s

t5

ru

t5 r

u

t5 rsangular speedd 5 6sangular speedd

400p radymin ≤ angular speed ≤ 1000p radymin

2ps200d ≤ angular speed ≤ 2ps500d

200 ≤revolutions

minute≤ 500

101. False, 1 radian so one radian

is much larger than one degree.

5 1180

p 28< 57.38,

103. True:2p

31

p

41

p

125

8p 1 3p 1 p

125 p 5 1808

102. No, is coterminal with 1808.212608

104. (a) An angle is in standard position when the origin is the vertex and the initial side

coincides with the positive -axis.

(b) A negative angle is generated by a clockwise rotation.

(c) Angles that have the same initial and terminal sides are coterminal angles.

(d) An obtuse angle is between and 1808.908

x

105. If is constant, the length of the arc is

proportional to the radius and

hence increasing.

ss 5 rud,u 106. Let A be the area of a circular sector of radius

and central angle Then

A

pr25

u

2p ⇒ A 5

1

2 r2u.

u.

r

107. square metersA 51

2r2u 5

1

2s10d2 ?

p

35

50

3p 108. Because

Hence, A 51

2r2u 5

1

2152112

152 5 90 ft2.

s 5 ru, u 512

15.

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109.

(a) Domain:

Domain:

The area function changes more rapidly for because

it is quadratic and the arc length function is linear.

(b) Domain:

Domain: 0 < u < 2p s 5 ru 5 10u

0

0

2

320

A

s

0 < u < 2p r 5 10 ⇒ A 512s102du 5 50u

r > 1

r > 0 s 5 ru 5 rs0.8d

r > 0 u 5 0.8 ⇒ A 512r2s0.8d 5 0.4r2

0

0

12

8

As

A 512r2u, s 5 ru

113.

−2 2 3 4 5 6

−2

−3

−4

1

2

3

4

x

y

115.

x

y

−2−3−4 1 2 3 4

−3

1

3

4

5

117.

x

y

−2−3−4 1 2 3 4

−2

−4

−5

1

2

3

110. If a fan of greater diameter is installed, the angular speed does not change.

111. Answers will vary. 112. Answers will vary.

114.

x

y

−2−3−4 1 2 3 4

−2

−3

−6

1

2

116.

x

y

−3−4−5 1 2 3

−2

−3

−4

1

2

3

4

118.

x

y

−2 1 2 3 5 6

−2

−3

−4

1

2

3

4

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Section 4.2 Trigonometric Functions: The Unit Circle

Section 4.2 Trigonometric Functions: The Unit Circle 281

2.

cot u 5x

y5

12y13

5y135

12

5

sec u 51

x5

1

12y135

13

12

csc u 51

y5

1

5y135

13

5

tan u 5y

x5

5y13

12y135

5

12

cos u 5 x 512

13

sin u 5 y 55

13

sx, yd 5 112

13,

5

132

4.

cot u 5x

y5

24y5

23y55

4

3

sec u 51

x5

1

24y55 2

5

4

csc u 51

y5

1

23y55 2

5

3

tan u 5y

x5

23y5

24y55

3

4

cos u 5 x 5 24

5

sin u 5 y 5 23

5

sx, yd 5 124

5, 2

3

52

6. t 5p

3 ⇒ 11

2, !3

2 2

1.

csc u 51

y5

17

15

sec u 51

x5 2

17

8

cot u 5x

y5 2

8

15

tan u 5y

x5 2

15

8

cos u 5 x 5 28

17

sin u 5 y 515

17

3.

csc u 51

y5 2

13

5

sec u 51

x5

13

12

cot u 5x

y5 2

12

5

tan u 5y

x5 2

5

12

cos u 5 x 512

13

sin u 5 y 5 25

13

Vocabulary Check

1. unit circle 2. periodic 3. odd, even

5. corresponds to 1!2

2, !2

2 2.t 5p

4

n You should know how to evaluate trigonometric functions using the unit circle.

n You should know the definition of a periodic function.

n You should be able to recognize even and odd trigonometric functions.

n You should be able to evaluate trigonometric functions with a calculator in both radian and degree mode.

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10. corresponds to 11

2, 2

!3

2 2.t 55p

3

12. t 5 p ⇒ s21, 0d

18. corresponds to

tan p

35

y

x5

!3y2

1y25 !3

cos p

35 x 5

1

2

sin p

35 y 5

!3

2

11

2, !3

2 2.t 5p

3


2, !3

2 2.t 52p

3

11. corresponds to s0, 21d.t 53p

2

13. corresponds to 1!2

2, !2

2 2.t 5 27p


1

2, !3

2 2.t 5 24p

3

15. corresponds to s0, 1d.t 5 23p

216. corresponds to s1, 0d.t 5 22p

17. corresponds to

tan t 5y

x5 1

cos t 5 x 5!2

2

sin t 5 y 5!2

2

1!2

2, !2

2 2.t 5p

4

19. corresponds to

tan t 5y

x5

1

!35

!3

3

cos t 5 x 5 2!3

2

sin t 5 y 5 21

2

12!3

2, 2

1

22.t 57p

620. corresponds to

tan t 5y

x5 21

cos t 5 x 5 2!2

2

sin t 5 y 5!2

2

12!2

2, !2

2 2.t 5 25p

4

21. corresponds to

tan t 5y

x5 2!3

cos t 5 x 5 21

2

sin t 5 y 5!3

2

121

2, !3

2 2.t 52p

322. corresponds to

tan t 5y

x5 2!3

cos t 5 x 51

2

sin t 5 y 5 2!3

2

11

2, 2

!3

2 2.t 55p

3

8. t 55p

4 ⇒ 12

!2

2, 2

!2

2 27. corresponds to 12!3

2, 2

1

22.t 57p

6

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24. corresponds to

tan t 5y

x5 2

!3

3

cos t 5 x 5!3

2

sin t 5 y 5 21

2

1!3

2, 2

1

22.t 511p

623. corresponds to

tan t 5y

x5 !3

cos t 5 x 51

2

sin t 5 y 5!3

2

11

2, !3

2 2.t 5 25p

3

25. corresponds to

tan t 5y

x5 2

!3

3

cos t 5 x 5!3

2

sin t 5 y 5 21

2

1!3

2, 2

1

22.t 5 2p

626. corresponds to

tan123p

4 2 5y

x5 1

cos123p

4 2 5 x 5 2!2

2

sin123p

4 2 5 y 5 2!2

2

12!2

2, 2

!2

2 2.t 5 23p

4

27. corresponds to

tan t 5y

x5 1

cos t 5 x 5!2

2

sin t 5 y 5!2

2

1!2

2, !2

2 2.t 5 27p

428. corresponds to

tan t 5y

x5 2!3

cos t 5 x 5 21

2

sin t 5 y 5!3

2

121

2, !3

2 2.t 5 24p

3

29. corresponds to

is undefined.tan t 5y

x

cos t 5 x 5 0

sin t 5 y 5 1

s0, 1d.t 5 23p

2

31. corresponds to

cot t 5x

y5 21tan t 5

y

x5 21

sec t 51

x5 2!2cos t 5 x 5 2

!2

2

csc t 51

y5 !2sin t 5 y 5

!2

2

12!2

2, !2

2 2 .t 53p

4

30. corresponds to

tans22pd 5y

x5

0

15 0

coss22pd 5 x 5 1

sins22pd 5 y 5 0

s1, 0d.t 5 22p

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ed.


32. corresponds to

cot t 5x

y5 2!3

sec t 51

x5 2

2

!35 2

2!3

3

csc t 51

y5 2

tan t 5y

x5 2

1

!35 2

!3

3

cos t 5 x 5 2!3

2

sin t 5 y 51

2

12!3

2,

1

22.t 55p

633. corresponds to

is undefined.

is undefined. cot t 5x

y5 0tan t 5

y

x

sec t 51

xcos t 5 x 5 0

csc t 51

y5 1sin t 5 y 5 1

s0, 1d.t 5p

2

35. corresponds to

cot u 5!3

3tan t 5

y

x5 !3

sec u 5 22cos t 5 x 5 21

2

csc t 5 22!3

3sin t 5 y 5 2

!3

2

121

2, 2

!3

2 2.t 5 22p

334. corresponds to

undefined

undefined

cot 3p

25

x

y5

0

215 0

sec 3p

25

1

x5

1

0 ⇒

csc 3p

25

1

y5

1

215 21

tan 3p

25

y

x5

21

0 ⇒

cos 3p

25 x 5 0

sin 3p

25 y 5 21

s0, 21d.t 53p

2

36. corresponds to

tan127p

4 2 5y

x5 1

cos127p

4 2 5 x 5!2

2

sin127p

4 2 5 y 5!2

2

1!2

2, !2

2 2.t 5 27p

4

cot127p

4 2 5x

y5 1

sec127p

4 2 51

x5 !2

csc127p

4 2 51

y5 !2

37. sin 5p 5 sin p 5 0

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39. cos 8p

35 cos

2p

35 2

1

238. Because

cos 7p 5 coss6p 1 pd 5 cos p 5 21

7p 5 6p 1 p :

40. Because

sin 9p

45 sin12p 1

p

42 5 sin p

45

!2

2

9p

45 2p 1

p

4:

42. Because

5 sin15p

6 2 51

2

sin1219p

6 2 5 sin124p 15p

6 2

219p

65 24p 1

5p

6:

44. Because :

5 cos 4p

35 2

1

2

cos128p

3 2 5 cos124p 14p

3 2

28p

35 24p 1

4p

3

41. cos1213p

6 2 5 cos12p

62 5 cos111p

6 2 5!3

2

43. sin129p

4 2 5 sin12p

42 5 2!2

2

45.

(a)

(b) cscs2td 5 2csc t 5 23

sins2td 5 2sin t 5 21

3

sin t 51

3

47.

(a)

(b) secs2td 51

coss2td5 25

cos t 5 coss2td 5 21

5

coss2td 5 21

546. cos

(a)

(b) secs2td 51

coss2td5

1

cos t5 2

4

3

coss2td 5 cos t 5 23

4

t 5 23

4

48.

(a)

(b) csc t 51

sinstd5

1

2sins2td5 2

8

3

sin t 5 2sins2td 5 23

8

sins2td 53

849.

(a)

(b) sinst 1 pd 5 2sin t 5 24

5

sinsp 2 td 5 sin t 54

5

sin t 54

5

50.

(a)

(b) cosst 1 pd 5 2cos t 5 24

5

cossp 2 td 5 2cos t 5 24

5

cos t 54

551. sin

7p

9< 0.6428

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56. cot 3.7 51

tan 3.7< 1.6007

52. tan 2p

5< 3.0777 53. cos

11p

5< 0.8090 54. sin

11p

9< 20.6428

55. csc 1.3 < 1.0378 57. coss21.7d < 20.1288

59. csc 0.8 51

sin 0.8< 1.3940

76.

(a)

(b)

ys0d 51

4e20 coss0d 5

1

4 foot

ystd 51

4e2t cos 6t

70. (a)

(b) cos 2.5 5 x < 20.8

sin 0.75 5 y < 0.7

72. (a)

(b)

t < 0.72 or t < 5.56

cos t 5 0.75

t < 4.0 or t < 5.4

sin t 5 20.75

58. coss22.5d < 20.8011 60. sec 1.8 51

cos 1.8< 24.4014

62. sins213.4d < 20.740461. sec 22.8 51

cos 22.8< 21.4486

69. (a)

(b) cos 2 < 20.4

sin 5 < 21

63. cot 2.5 51

tan 2.5< 21.3386

64. tan 1.75 < 25.5204 65. cscs21.5d 51

sins21.5d < 21.0025

66. tans22.25d < 1.2386 67. secs24.5d 51

coss24.5d < 24.7439

68. cscs25.2d 51

sins25.2d < 1.1319

71. (a)

(b)

t < 1.82 or 4.46

cos t 5 20.25

t < 0.25 or 2.89

sin t 5 0.25 73.

amperes < 0.79

Is0.7d 5 5e21.4 sin 0.7

I 5 5e22t sin t

74. At

I < 5e22s1.4d sin 1.4 < 0.30 amperes.

t 5 1.4, 75.

(a)

(b)

(c) ys12d 5

14 cos 3 < 20.2475 ft

ys14d 5

14 cos

32 < 0.0177 ft

ys0d 514 cos 0 5 0.2500 ft

ystd 514 cos 6t

(c) The maximum displacements are decreasing

because of friction, which is modeled by the

term.

(d)

t 5p

12,

p

4

6t 5p

2,

3p

2

cos 6t 5 0

1

4e2t cos 6t 5 0

e2t

0.50 1.02 1.54 2.07 2.59

0.09 0.03 20.0220.0520.15y

t

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78. True

80. True

82. (a) The points and are symmetric

about the origin.

(b) Because of the symmetry of the points, you can

make the conjecture that

(c) Because of the symmetry of the points, you can

make the conjecture that cosst1 1 pd 5 2cos t1.

sinst1 1 pd 5 2sin t1.

sx2, y2dsx1, y1d

84.

Therefore, sin t1 1 sin t2 Þ sinst1 1 t2d.

sin 1 < 0.8415

sins0.25d 1 sins0.75d < 0.2474 1 0.6816 5 0.9290

77. False. sin124p

3 2 5!3

2> 0

79. False. 0 corresponds to s1, 0d.

81. (a) The points have -axis symmetry.

(b) since they have the same

-value.

(c) since the -values have

opposite signs.

x2cos t1 5 cos sp 2 t1d

y

sin t1 5 sin sp 2 t1d

y

83.

Thus, cos 2t Þ 2 cos t.

2 cos 0.75 < 1.4634cos 1.5 < 0.0707,

85.

θ

θ−

(x, y)

(x, −y)

x

y

sec u 51

cos u5

1

coss2ud 5 secs2ud

cos u 5 x 5 coss2ud

87. is odd.

hs2td 5 f s2tdgs2td 5 2f stdgstd 5 2hstd

hstd 5 f stdgstd 88. and

Both and are odd functions.

The function is even.hstd 5 fstdgstd

5 sin t tan t 5 hstd

5 s2sin tds2tan td

hs2td 5 sins2td tans2td

hstd 5 fstdgstd 5 sin t tan t

gf

gstd 5 tan tfstd 5 sin t

86.

cot u 5x

y5 2cots2ud

tan u 5y

x5 2tans2ud

csc u 51

y5 2cscs2ud

sin u 5 y 5 2sins2ud

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90.

f21sxd 5 3!4sx 2 1d

y 5 3!4sx 2 1d

4sx 2 1d 5 y3

x 2 1 514 y3

x 514 y3 1 1

y 514x3 1 1

−9 9

−6

6

ff−1

fsxd 514 x3 1 189.

f 21sxd 523sx 1 1d

23sx 1 1d 5 y

2x 1 2 5 3y

2x 5 3y 2 2

x 512s3y 2 2d

y 512s3x 2 2d

−9 9

−6

6

ff−1

f sxd 512s3x 2 2d

92.

f21sxd 5x

2 2 x, x < 2

x

2 2 x5 y, x < 2

x 5 ys2 2 xd

x 5 2y 2 xy

xy 1 x 5 2y

x 52y

y 1 1

y 52x

x 1 1, x > 21

−3 9

−4

4

fsxd 52x

x 1 1, x > 2191.

f21sxd 5 !x2 1 4, x ≥ 0

!x2 1 4 5 y, x ≥ 0

x2 1 4 5 y2

x2 5 y2 2 4

x 5 !y2 2 4

−2 10

−1

7

ff−1

y 5 !x2 2 4

f sxd 5 !x2 2 4, x ≥ 2, y ≥ 0

94.

Asymptotes:

10864

4

6

8

10

−6

−8

x

y

x 5 23, x 5 2, y 5 0

f sxd 55x

x2 1 x 2 65

5x

sx 1 3dsx 2 2d93.

Asymptotes:

2 4 5 6 7 8−2 −1

1

3

4

5

6

7

8

−2

−3

−4

−3−4x

y

x 5 3, y 5 2

f sxd 52x

x 2 3

95.

Asymptotes: x 5 22, y 51

2

x Þ 2 5x 1 5

2sx 1 2d,

f sxd 5x2 1 3x 2 10

2x2 2 85

sx 2 2dsx 1 5d2sx 2 2dsx 1 2d

x

y

−1−5−6−7 1−1

−2

−3

−4

1

2

3

4

−2

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Section 4.3 Right Triangle Trigonometry 289

96.

Slant asymptote:

Vertical asymptotes: x < 3.608, x < 21.108

y 5x

22

7

4 4

6

8

10

−8

−10

−2−4−6−8 6 8 10

2

y

x

f sxd 5x3 2 6x2 1 x 2 1

2x2 2 5x 2 85

x

22

7

42

15sx 1 4d4s2x2 2 5x 2 8d

97.

Domain: all real numbers

Intercepts:

No asymptotes

s0, 24d, s1, 0d, s24, 0d

0 5 x2 1 3x 2 4 5 sx 1 4dsx 2 1d ⇒ x 5 1, 24

y 5 x2 1 3x 2 4 98.

Domain: all

Intercepts:

Asymptote: x 5 0

s1, 0d, s21, 0d

ln x4 5 0 ⇒ x4 5 1 ⇒ x 5 ±1

x Þ 0

y 5 ln x4

99.


Intercept:

Asymptote: y 5 2

s0, 5d

fsxd 5 3x11 1 2 100.


Intercepts:

Asymptotes: x 5 22, y 5 0

s7, 0d, 10, 27

42x Þ 22

fsxd 5x 2 7

sx2 1 4x 1 4d 5x 2 7

sx 1 2d2

Section 4.3 Right Triangle Trigonometry

n You should know the right triangle definition of trigonometric functions.

(a) (b) (c)

(d) (e) (f)

n You should know the following identities.

(a) (b) (c)

(d) (e) (f)

(g) (h) (i)

( j) (k)

n You should know that two acute angles are complementary if and cofunctions of

complementary angles are equal.

n You should know the trigonometric function values of and or be able to construct triangles from

which you can determine them.

608,308, 458,

a 1 b 5 908,a and b

1 1 cot2 u 5 csc2 u1 1 tan2 u 5 sec2 u

sin2 u 1 cos2 u 5 1cot u 5cos u

sin utan u 5

sin u

cos u

cot u 51

tan utan u 5

1

cot usec u 5

1

cos u

cos u 51

sec ucsc u 5

1

sin usin u 5

1

csc u

cot u 5adj

oppsec u 5

hyp

adjcsc u 5

hyp

opp

Adjacent side

Opposi

te s

ide

Hyp

oten

use

θ

tan u 5opp

adjcos u 5

adj

hypsin u 5

opp

hyp

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Vocabulary Check

1. (a) iii (b) vi (c) ii (d) v (e) i (f) iv

2. hypotenuse, opposite, adjacent 3. elevation, depression

1.

cot u 5adj

opp5

4

3

sec u 5hyp

adj5

5

4

csc u 5hyp

opp5

5

3

tan u 5opp

adj5

3

4

cos u 5adj

hyp5

4

5

sin u 5opp

hyp5

3

5

adj 5 !52 2 32 5 !16 5 4

35

θ

3.

cot u 5adj

opp5

15

8

sec u 5hyp

adj5

17

15

csc u 5hyp

opp5

17

8

tan u 5opp

adj5

8

15

cos u 5adj

hyp5

15

17

sin u 5opp

hyp5

8

17

hyp 5 !82 1 152 5 17

θ

15

8

2.

sin

cos

tan

csc

sec

cot u 5adj

opp5

12

5

u 5hyp

adj5

13

12

u 5hyp

opp5

13

5

u 5opp

adj5

5

12

u 5adj

hyp5

12

13

u 5opp

hyp5

5

13

b 5 !132 2 52 5 !169 2 25 5 12

513

b

θ

4.

sin

cos

tan

csc

sec

cot u 52

3

u 5!13

2

u 5!13

3

u 518

125

3

2

u 512

6!135

2!13

13

u 518

6!135

3!13

13

C 5 !182 1 122 5 !468 5 6!13

12

18 c

θ

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6.

sin

cos

tan

csc

sec

cot u 5adj

opp5

!161

8

u 5hyp

adj5

15

!1615

15!161

161

u 5hyp

opp5

15

8

u 5opp

adj5

8

!1615

8!161

161

u 5adj

hyp5

!161

15

u 5opp

hyp5

8

15

158

θ

adj 5 !152 2 82 5 !161

sin

cos

tan

csc

sec

cot u 5adj

opp5

!161

2 ? 45

!161

8

u 5hyp

adj5

7.5

s!161y2d 515

!1615

15!161

161

u 5hyp

opp5

7.5

45

15

8

u 5opp

adj5

4

s!161y2d 58

!1615

8!161

161

u 5adj

hyp5

!161

2 ? 7.55

!161

15

u 5opp

hyp5

4

7.55

8

15

7.54

θ

adj 5 !7.52 2 42 5!161

2

The function values are the same because the triangles are similar, and corresponding sides are proportional.

5.

cot u 5adj

opp5

8

65

4

3

sec u 5hyp

adj5

10

85

5

4

csc u 5hyp

opp5

10

65

5

3

tan u 5opp

adj5

6

85

3

4

cos u 5adj

hyp5

8

105

4

5

sin u 5opp

hyp5

6

105

3

5

θ

8

10

opp 5 !102 2 82 5 6

cot u 5adj

opp5

2

1.55

4

3

sec u 5hyp

adj5

2.5

25

5

4

csc u 5hyp

opp5

2.5

1.55

5

3

tan u 5opp

adj5

1.5

25

3

4

cos u 5adj

hyp5

2

2.55

4

5

sin u 5opp

hyp5

1.5

2.55

3

5

θ

2

2.5opp 5 !2.52 2 22 5 1.5

The function values are the same since the triangles are similar and the corresponding sides are proportional.

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7.

cot u 5adj

opp5 2!2

sec u 5hyp

adj5

3

2!25

3!2

4

csc u 5hyp

opp5 3

tan u 5opp

adj5

1

2!25

!2

4

cos u 5adj

hyp5

2!2

3

θ3

1sin u 5

opp

hyp5

1

3

adj 5 !32 2 12 5 !8 5 2!2

8.

sin

cos

tan

csc

sec

cot u 5adj

opp5

2

15 2

u 5hyp

adj5

!5

2

u 5hyp

opp5

!5

15 !5

u 5opp

adj5

1

2

u 5adj

hyp5

2

!55

2!5

5

u 5opp

hyp5

1

!55

!5

5

1

2

θhyp 5 !12 1 22 5 !5

sin

cos

tan

csc

sec

cot u 56

35 2

u 53!5

65

!5

2

u 53!5

35 !5

u 53

65

1

2

u 56

3!55

2

!55

2!5

5

u 53

3!55

1

!55

!5

5

3

6

θ

hyp 5 !32 1 62 5 3!5

The function values are the same because the triangles are similar, and corresponding sides are proportional.

cot u 5adj

opp5

4!2

25 2!2

sec u 5hyp

adj5

6

4!25

3

2!25

3!2

4

csc u 5hyp

opp5

6

25 3

tan u 5opp

adj5

2

4!25

1

2!25

!2

4

cos u 5adj

hyp5

4!2

65

2!2

3

θ

62

sin u 5opp

hyp5

2

65

1

3

adj 5 !62 2 22 5 !32 5 4!2

The function values are the same since the triangles are similar and the corresponding sides are proportional.

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10.

sin

cos

tan

csc

sec u 5hyp

adj5

!26

5

u 5hyp

opp5

!26

15 !26

u 5opp

adj5

1

5

u 5adj

hyp5

5

!265

5!26

26

u 5opp

hyp5

1

!265

!26

26

1

5

26θ

hyp 5 !52 1 12 5 !26

12.

sin

tan

csc

sec

cot u 53

2!105

3!10

20

u 57

3

u 57

2!105

7!10

20

u 52!10

3

u 52!10

7

θ

102

3

7

opp 5 !72 2 32 5 !40 5 2!10

9. Given:

csc u 5hyp

opp5

6

5

sec u 5hyp

adj5

6

!115

6!11

11

cot u 5adj

opp5

!11

5

tan u 5opp

adj5

5

!115

5!11

11

cos u 5adj

hyp5

!11

6

adj 5 !11

52 1 sadjd2 5 62

56

11

θ

sin u 55

65

opp

hyp

11. Given:

csc u 54

!155

4!15

15

cot u 51

!155

!15

15

tan u 5 !15

cos u 51

4

sin u 5!15

4

opp 5 !15

soppd2 1 12 5 42

15

θ

4

1

sec u 5 4 54

15

hyp

adj

14.

sin

cos

tan

sec

cot u 51

tan u5

!273

4

u 51

cos u5

17

!2735

17!273

273

u 5opp

adj5

4

!2735

4!273

273

u 5adj

hyp5

!273

17

u 5opp

hyp5

4

17

417

273

θ

adj 5 !172 2 42 5 !27313. Given:

csc u 5!10

3

cot u 51

3

sec u 5 !10

cos u 5!10

10

sin u 53!10

10

hyp 5 !10

32 1 12 5 shypd2

3

1

10

θ

tan u 5 3 53

15

opp

adj

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294 Chapter 4 Trigonometric Functions294 Chapter 4 Trigonometric Functions

16.

cos

tan

csc

sec

cot u 51

tan u5

!55

3

u 51

cos u5

8

!555

8!55

55

u 51

sin u5

8

3

u 5opp

adj5

3

!555

3!55

55

u 5adj

hyp5

!55

8

adj 5 !82 2 32 5 !55

38

55

θ

sin u 53

815. Given:

csc u 5!97

4

sec u 5!97

9

tan u 54

9

cos u 59

!975

9!97

97

sin u 54

!975

4!97

97

hyp 5 !97

42 1 92 5 shypd2

97

θ

9

4

cot u 59

45

adj

opp

Function (deg) (rad) Function Valueuu

26. tan1

!3

p

6308

24. sin!2

2

p

4458

22. csc !2p

4458

20. sec !2p

4458

18. cos!2

2

p

4458

Function (deg) (rad) Function Value

17. sin

19. tan

21. cot

23. cos

25. cot 1p

4458

!3

2

p

6308

!3

3

p

3608

!3p

3608

1

2

p

6308

uu

27. sin u 51

csc u28. cos u 5

1

sec u29. tan u 5

1

cot u30. csc u 5

1

sin u

31. sec u 51

cos u32. cot u 5

1

tan u 33. tan u 5sin u

cos u34. cot u 5

cos u

sin u

35. sin2 u 1 cos2 u 5 1 36. 1 1 tan2 u 5 sec2 u 37. sins908 2 ud 5 cos u 38. coss908 2 ud 5 sin u

39. tans908 2 ud 5 cot u 40. cots908 2 ud 5 tan u 41. secs908 2 ud 5 csc u 42. cscs908 2 ud 5 sec u

43.

(a)

(c) cos 308 5 sin 608 5!3

2

tan 608 5sin 608

cos 6085 !3

sin 608 5!3

2, cos 608 5

1

2

(b)

(d) cot 608 5cos 608

sin 6085

1

!35

!3

3

sin 308 5 cos 608 51

2

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44.

(a)

(b)

(c)

(d) cot 308 51

tan 3085

3

!35

3!3

35 !3

cos 308 5sin 308

tan 3085

s1y2ds!3y3d 5

3

2!35

!3

2

cot 608 5 tans908 2 608d 5 tan 308 5!3

3

csc 308 51

sin 3085 2

sin 308 51

2, tan 308 5

!3

3

46.

(a)

(b) cot u 51

tan u5

1

2!65

!6

12

cos u 51

sec u5

1

5

sec u 5 5, tan u 5 2!6

45.

(a)

(b)

(c)

(d) secs90º 2 ud 5 csc u 5 3

tan u 5sin u

cos u5

1y3

s2!2 dy35

!2

4

cos u 51

sec u5

2!2

3

sin u 51

csc u5

1

3

csc u 5 3, sec u 53!2

4

47.

(a)

(b)

sin a 5!15

4

sin2 a 515

16

sin2 a 1 11

422

5 1

sin2 a 1 cos2 a 5 1

sec a 51

cos a5 4

cos a 51

4

(c)

(d) sins908 2 ad 5 cos a 51

4

5!15

15 5

1

!15

51y4

!15y4

cot a 5cos a

sin a

(c)

(d) sin u 5 tan u cos u 5 s2!6 d11

52 52!6

5

cots908 2 ud 5 tan u 5 2!6

48.

(a)

(b)

(c)

(d) csc b 5 !1 1 cot2 b 5!1 11

255

!26

5

tans908 2 bd 5 cot b 51

5

sec2 b 5 1 1 tan2 b ⇒ cos b 5 1

!1 1 tan2 b5

1

!1 1 255

1

!265

!26

26

cot b 51

tan b5

1

5

sb lies in Quadrant I or III.dtan b 5 5

50. csc u tan u 51

sin u?

sin u

cos u5

1

cos u5 sec u49. tan u cot u 5 tan u1 1

tan u2 5 1

51. tan u cos u 5 1 sin u

cos u2 cos u 5 sin u 52. cot u sin u 5cos u

sin u sin u 5 cos u

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53.

5 sin2 u

5 ssin2 u 1 cos2 ud 2 cos2 u

s1 1 cos uds1 2 cos ud 5 1 2 cos2 u

55.

5 csc u sec u 51

sin u?

1

cos u

51

sin u cos u

sin u

cos u1

cos u

sin u5

sin2 u 1 cos2 u

sin u cos u

54.

5 1

scsc u 1 cot udscsc u 2 cot ud 5 csc2 u 2 cot2 u

56.

5 1 1 cot2 u 5 csc2 u

5 1 1cot u

s1ycot ud

tan u 1 cot u

tan u5

tan u

tan u1

cot u

tan u

57. (a)

(b) cos 878 < 0.0523

sin 418 < 0.6561

59. (a)

(b) csc 48º 79 51

sins48 1760dº < 1.3432

sec 428 129 5 sec 42.28 51

cos 42.28< 1.3499

58. (a)

(b) cot 71.58 51

tan 71.58< 0.3346

tan 18.58 < 0.3346

60. (a)

(b) secs88 509 250 d 51

coss88 509 250 d < 1.0120

< coss8.840278d < 0.9881

coss88 509 250 d 5 cos18 150

601

25

36002

61. Make sure that your calculator is in radian mode.

(a)

(b) tan p

8< 0.4142

cot p

165

1

tanspy16d< 5.0273

63. (a)

(b) csc u 5 2 ⇒ u 5 308 5p

6

sin u 51

2 ⇒ u 5 308 5

p

6

62. (a)

(b) coss1.25d < 0.3153

secs1.54d 51

coss1.54d < 32.4765

64. (a)

(b) tan u 5 1 ⇒ u 5 458 5p

4

cos u 5!2

2 ⇒ u 5 458 5

p

4

65. (a)

(b) cot u 5 1 ⇒ u 5 458 5p

4

sec u 5 2 ⇒ u 5 608 5p

3

67. (a)

(b) sin u 5!2

2 ⇒ u 5 458 5

p

4

csc u 52!3

3 ⇒ u 5 608 5

p

3

66. (a)

(b) cos u 51

2 ⇒ u 5 608 5

p

3

tan u 5 !3 ⇒ u 5 608 5p

3

68. (a)

(b)

cos u 51

!25

!2

2 ⇒ u 5 458 5

p

4

sec u 5 !2

tan u 53

!35 !3 ⇒ u 5 608 5

p

3

cot u 5!3

3

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69.

cos 308 5105

r ⇒ r 5

105

cos 3085

105

!3y25

210

!35 70!3

tan 308 5y

105 ⇒ y 5 105 tan 308 5 105 ?

!3

35 35!3

70.

sin 308 5y

15 ⇒ y 5 15 sin 308 5 1511

22 515

2

cos 308 5x

15 ⇒ x 5 15 cos 308 5 15 ?

!3

25

15!3

2

71.

sin 608 5y

16 ⇒ y 5 16 sin 608 5 161!3

2 2 5 8!3

cos 608 5x

16 ⇒ x 5 16 cos 608 5 1611

22 5 8

72.

sin 608 538

r ⇒ r 5

38

sin 6085

38

!3y25

76!3

3

cot 608 5x

38 ⇒ x 5 38 cot 608 5 38 ?

1

!35

38!3

3

73.

r2 5 202 1 202 ⇒ r 5 20!2

tan 458 520

x ⇒ 1 5

20

x ⇒ x 5 20 74.

r2 5 102 1 102 ⇒ r 5 10!2

tan 458 5y

10 ⇒ 1 5

y

10 ⇒ y 5 10

75.

r2 5 s2!5d2 1 s2!5d2 5 20 1 20 5 40 ⇒ r 5 2!10

tan 458 52!5

x ⇒ 1 5

2!5

x ⇒ x 5 2!5

76.

r2 5 s4!6d2 1 s4!6d2 5 96 1 96 5 192 ⇒ r 5 8!3

tan 458 5y

4!6 ⇒ 1 5

y

4!6 ⇒ y 5 4!6

77. (a)

(b) and Thus,

(c) feeth 56s21d

55 25.2

6

55

h

21.tan u 5

h

21tan u 5

6

5

6

h

516

θ

78. (a)

(b)

(c) x 5 30 sin 758 < 28.98 meters

sin 758 5x

30

x

75°

30

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80.

Subtracting,

< 1.295 < 1.3 miles. <13

16.3499 2 6.3138

h 513

cot 3.58 2 cot 98

13

h5 cot 3.58 2 cot 98

cot 3.58 513 1 c

h

not drawn to scale

9°3.5°

13 c

h

cot 98 5c

h79.

feetw 5 100 tan 588 < 160

tan 588 5w

100

tan u 5opp

adj

81. (a)

(b)

(c) rate down the zip line

vertical rate50

65

25

3 ftysec

50!2

65

25!2

3 ftysec

L2 5 502 1 502 5 2 ? 502 ⇒ L 5 50!2 feet

tan u 550

505 1 ⇒ u 5 458 82. (a)

(b) feet above sea level

(c) minutes to reach top

Vertical rate

feet per minute < 173.8

5519.3

s896.5y300d

896.5

300

1693.5 2 519.3 5 1174.2

x 5 896.5 sin 35.48 < 519.3 feet

sin 35.48 5x

896.5

83.

sx1, y1d 5 s28!3, 28d

x1 5 cos 308s56d 5!3

2s56d 5 28!3

cos 308 5x1

56

y1 5 ssin 308ds56d 5 11

22s56d 5 28

sin 308 5y1

56

30°

56

( , )x y1 1

sx2, y2d 5 s28, 28!3 d

x2 5 scos 608ds56d 5 11

22s56d 5 28

cos 608 5x2

56

y2 5 sin 608s56d 5 1!3

2 2s56d 5 28!3

sin 60º 5y2

56

60°

56

( , )x y2 2

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84.

csc 208 510

y< 2.92sec 208 5

10

x< 1.06

cot 208 5x

y< 2.75tan 208 5

y

x< 0.36

cos 208 5x

10< 0.94sin 208 5

y

10< 0.34

x < 9.397, y < 3.420

86. False

5 !2 Þ 1 sin 458 1 cos 458 5!2

21

!2

2

87. True

for all u1 1 cot2 u 5 csc2 u

85. True

sin 608 csc 608 5 sin 608 1

sin 6085 1

88. No. so you can find ± sec u.tan2 u 1 1 5 sec2 u,

(b) Sine and tangent are increasing, cosine is

decreasing.

(c) In each case, tan u 5sin u

cos u.

89. (a)

0 0.3420 0.6428 0.8660 0.9848

1 0.9397 0.7660 0.5000 0.1736

0 0.3640 0.8391 1.7321 5.6713tan u

cos u

sin u

80860840820808u

90.

1 0.9397 0.7660 0.5000 0.1736

1 0.9397 0.7660 0.5000 0.1736sins908 2 ud

cos u

80860840820808u

are complementary angles.u and 908 2 u

cos u 5 sins908 2 ud

91.

−6 6

−1

7 93.

−6 6

−1

7

95.

−2 10

−4

4 97.

−2 10

−4

4

92.

−3 3

−3

1 94.

−6 6

−6

2

96.

−2 10

−4

4 98.

−2 10

−4

4

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Section 4.4 Trigonometric Functions of Any Angle


n Know the Definitions of Trigonometric Functions of Any Angle.

If is in standard position, a point on the terminal side and then:

n You should know the signs of the trigonometric functions in each quadrant.

n You should know the trigonometric function values of the quadrant angles

n You should be able to find reference angles.

n You should be able to evaluate trigonometric functions of any angle. (Use reference angles.)

n You should know that the period of sine and cosine is

n You should know which trigonometric functions are odd and even.

Even: cos x and sec x

Odd: sin x, tan x, cot x, csc x

2p.

0, p

2, p, and

3p

2.

cot u 5x

y, y Þ 0tan u 5

y

x, x Þ 0

sec u 5r

x, x Þ 0cos u 5

x

r

csc u 5r

y, y Þ 0sin u 5

y

r

r 5 !x2 1 y2 Þ 0,sx, ydu

Vocabulary Check

1. 2. 3. 4.

5. 6. 7. referencecot ucos u

r

x

y

xcsc u

y

r

1. (a)

cot u 5x

y5

4

3tan u 5

y

x5

3

4

sec u 5r

x5

5

4cos u 5

x

r5

4

5

csc u 5r

y5

5

3sin u 5

y

r5

3

5

r 5 !16 1 9 5 5

sx, yd 5 s4, 3d (b)

cot u 5x

y5

8

15tan u 5

y

x5

15

8

sec u 5r

x5 2

17

8cos u 5

x

r5 2

8

17

csc u 5r

y5 2

17

15sin u 5

y

r5 2

15

17

r 5 !64 1 225 5 17

sx, yd 5 s28, 215d

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Section 4.4 Trigonometric Functions of Any Angle 301

2. (a)

cot u 5x

y5

12

255 2

12

5

sec u 5r

x5

13

12

csc u 5r

y5

13

255 2

13

5

tan u 5y

x5

25

125 2

5

12

cos u 5x

r5

12

13

sin u 5y

r5

25

135 2

5

13

r 5 !122 1 s25d2 5 13

x 5 12, y 5 25 (b)

cot u 5x

y5

21

15 21

sec u 5r

x5

!2

215 2!2

csc u 5r

y5

!2

15 !2

tan u 5y

x5

1

215 21

cos u 5x

r5

21

!25 2

!2

2

sin u 5y

r5

1

!25

!2

2

r 5 !s21d2 1 12 5 !2

x 5 21, y 5 1

4. (a)

cot u 5x

y5

3

15 3

sec u 5r

x5

!10

3

csc u 5r

y5

!10

15 !10

tan u 5y

x5

1

3

cos u 5x

r5

3

!105

3!10

10

sin u 5y

r5

1

!105

!10

10

r 5 !32 1 12 5 !10

x 5 3, y 5 1 (b)

cot u 5x

y5

2

245 2

1

2

sec u 5r

x5

2!5

25 !5

csc u 5r

y5

2!5

245 2

!5

2

tan u 5y

x5

24

25 22

cos u 5x

r5

2

2!55

!5

5

sin u 5y

r5

24

2!55 2

2!5

5

r 5 !22 1 s24d2 5 2!5

x 5 2, y 5 24

3. (a)

cot u 5x

y5 !3tan u 5

y

x5

!3

3

sec u 5r

x5

22!3

3cos u 5

x

r5

2!3

2

csc u 5r

y5 22sin u 5

y

r5 2

1

2

r 5 !3 1 1 5 2

sx, yd 5 s2!3, 21d (b)

cot u 5x

y5 21tan u 5

y

x5 21

sec u 5r

x5 2!2cos u 5

x

r5 2

!2

2

csc u 5r

y5 !2sin u 5

y

r5

!2

2

r 5 !4 1 4 5 2!2

sx, yd 5 s22, 2d

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5.

cot u 5x

y5

7

24tan u 5

y

x5

24

7

sec u 5r

x5

25

7cos u 5

x

r5

7

25

csc u 5r

y5

25

24sin u 5

y

r5

24

25

r 5 !49 1 576 5 25

sx, yd 5 s7, 24d

7.

cot u 5x

y5 2

5

12

sec u 5r

x5

13

5

csc u 5r

y5 2

13

12

tan u 5y

x5 2

12

5

cos u 5x

r5

5

13

sin u 5y

r5 2

12

13

r 5 !52 1 s212d2 5 !25 1 144 5 !169 5 13

sx, yd 5 s5, 212d

6.

sin cos

tan csc

sec cot u 5x

y5

8

15u 5

r

x5

17

8

u 5r

y5

17

15u 5

y

x5

15

8

u 5x

r5

8

17u 5

y

r5

15

17

r 5 !82 1 152 5 17

x 5 8, y 5 15,

8.

sin

cos

tan

csc

sec

cot u 5x

y5 2

12

5

u 5r

x5 2

13

12

u 5r

y5

13

5

u 5y

x5

10

2245 2

5

12

u 5x

r5

224

265

212

13

u 5y

r5

10

265

5

13

r 5 !s224d2 1 s10d2 5 26

x 5 224, y 5 10

9.

cot u 5x

y5 2

2

5

sec u 5r

x5 2

!29

2

csc u 5r

y5

!29

5

tan u 5y

x5 2

5

2

cos u 5x

r5 2

2!29

29

sin u 5y

r5

5!29

29

r 5 !16 1 100 5 2!29

sx, yd 5 s24, 10d 10.

sin

cos

tan

csc

sec

cot u 5x

y5

5

6

u 5r

x5 2

!61

5

u 5r

y5 2

!61

6

u 5y

x5

26

255

6

5

u 5x

r5

25

!615

25!61

61

u 5y

r5

26

!615

26!61

61

r 5 !s25d22 1 s26d2 5 !61

x 5 25, y 5 26

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11.

cot u 5x

y5 2

5

4

sec u 5r

x5 2

2!41

5

csc u 5r

y5

!41

4

tan u 5y

x5

8

2105 2

4

5

cos u 5x

r5

210

2!415 2

5!41

41

sin u 5y

r5

8

2!415

4!41

41

r 5 !s210d2 1 82 5 !164 5 2!41

sx, yd 5 s210, 8d 12.

sin

cos

tan

csc

sec

cot u 5x

y5 2

1

3

u 5r

x5 !10

u 5r

y5 2

!10

3

u 5y

x5

29

35 23

u 5x

r5

3

3!105

!10

10

u 5y

r5

29

3!105

23!10

10

r 5 !32 1 s29d2 5 !90 5 3!10

x 5 3, y 5 29

13. lies in Quadrant III or Quadrant IV.

lies in Quadrant II or Quadrant III.

and lies in Quadrant III.cos u < 0 ⇒ usin u < 0

cos u < 0 ⇒ u

sin u < 0 ⇒ u

15. lies in Quadrant I or Quadrant III.

lies in Quadrant I or Quadrant IV.

and lies in Quadrant I.cos u > 0 ⇒ ucot u > 0

cos u > 0 ⇒ u

cot u > 0 ⇒ u

17.

in Quadrant II

cot u 5x

y5 2

4

3tan u 5

y

x5 2

3

4

sec u 5r

x5 2

5

4cos u 5

x

r5 2

4

5

csc u 5r

y5

5

3sin u 5

y

r5

3

5

⇒ x 5 24u

sin u 5y

r5

3

5 ⇒ x2 5 25 2 9 5 16

14. and

and

Quadrant IV

x

y< 0

r

x> 0

cot u < 0sec u > 0

16. and

and

Quadrant III

r

y< 0

y

x> 0

csc u < 0tan u > 0

18.

in Quadrant III

sin csc

cos sec

tan cot u 54

3u 5

y

x5

3

4

u 5 25

4u 5

x

r5 2

4

5

u 5 25

3u 5

y

r5 2

3

5

⇒ y 5 23u

cos u 5x

r5

24

5 ⇒ |y| 5 3

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19.

cot u 5x

y5 2

8

15

sec u 5r

x5

17

8cos u 5

x

r5

8

17

csc u 5r

y5 2

17

15sin u 5

y

r5 2

15

17

tan u 5y

x5

215

8 ⇒ r 5 17

sin u < 0 ⇒ y < 0 20. csc

cot

sin csc

cos sec

tan cot u 5 2!15u 5y

x5 2

!15

15

u 5 24!15

15u 5

x

r5 2

!15

4

u 5 4u 5y

r5

1

4

u < 0 ⇒ x 5 2!15

u 5r

y5

4

1 ⇒ x 5 ±!15

21.

cot u 5x

y5 2

!3

3tan u 5

y

x5 2!3

sec u 5r

x5 22cos u 5

x

r5 2

1

2

csc u 5r

y5

2!3

3sin u 5

y

r5

!3

2

sin u ≥ 0 ⇒ y 5 !3

sec u 5r

x5

2

21 ⇒ y2 5 4 2 1 5 3

23. is undefined

is undefined.

is undefined.cot utan u 5y

x5

0

x5 0

sec u 5r

x5 21cos u 5

x

r5

2r

r5 21

csc u 5r

ysin u 5

y

r5 0

p

2≤ u ≤

3p

2⇒ u 5 p, y 5 0, x 5 2r

⇒ u 5 np.cot u

22. and

is undefined.

is undefined.cot u

sec u 5 21

csc u

tan u 5sin u

cos u5 0

cos u 5 cos p 5 21

p

2≤ u ≤

3p

2 ⇒ u 5 psin u 5 0

24. is undefined and

is undefined.

is undefined.

cot u 5 0

sec u

csc u 5 21

tan u

cos u 5 0

sin u 5 21

p ≤ u ≤ 2p ⇒ u 53p

2.tan u

25. To find a point on the terminal side of use any point on the line that lies in

Quadrant II. is one such point.

cot u 5 21tan u 5 21

sec u 5 2!2cos u 5 21

!25 2

!2

2

csc u 5 !2sin u 51

!25

!2

2

x 5 21, y 5 1, r 5 !2

s21, 1dy 5 2xu,

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27. To find a point on the terminal side of use any

point on the line that lies in Quadrant III.

is one such point.

cot u 521

225

1

2

sec u 5!5

215 2!5

csc u 5!5

225 2

!5

2

tan u 522

215 2

cos u 5 21

!55 2

!5

5

sin u 5 22

!55 2

2!5

5

x 5 21, y 5 22, r 5 !5

s21, 22dy 5 2x

u,26. Quadrant III,

sin

cos

tan

csc

sec

cot u 5x

y5

2x

s21y3d x5 3

u 5r

x5

s!10xdy3

2x5 2

!10

3

u 5r

y5

s!10xdy3

s21y3dx5 2!10

u 5y

x5

s21y3dx

2x5

1

3

u 5x

r5

2x

s!10xdy35 2

3!10

10

u 5y

r5

s2xy3ds!10xdy3

5 2!10

10

r 5!x2 11

9x2 5

!10x

3

x > 012x, 21

3x2

28.

Quadrant IV,

sin csc

cos sec

tan cot u 5 23

4u 5

y

x5

s24y3d x

x5 2

4

3

u 55

3u 5

x

r5

x

s5y3dx5

3

5

u 5 25

4u 5

y

r5

s24y3dx

s5y3d x5 2

4

5

r 5!x2 116

9x2 5

5

3x

x > 01x, 24

3x2

4x 1 3y 5 0 ⇒ y 5 24

3x

29.

sec p 5r

x5

1

215 21

sx, yd 5 s21, 0d 31.

50

215 0 cot13p

2 2 5x

y

sx, yd 5 s0, 21d

33.

sec 0 5r

x5

1

15 1

sx, yd 5 s1, 0d

35.

undefined cot p 5x

y5 2

1

0 ⇒

sx, yd 5 s21, 0d

30. undefined

since corresponds to s0, 1d.p

2

tan p

25

y

x5

1

0 ⇒

32. undefinedcsc 0 5r

y5

1

0 ⇒ 34. csc

3p

25

1

sin 3p

2

51

215 21

36. csc p

25

1

sin p

2

51

15 1

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37.

u9 5 1808 2 1208 5 608

x

y

1208=θ608=θ ′

u 5 1208

39. is coterminal with

x

y

−2358=θ

558=θ ′

u9 5 2258 2 1808 5 458

2258.u 5 21358

38.

x

y

2258=θ

458=θ ′

u9 5 2258 2 1808 5 458

40.

x

y

−3308=θ 308=θ ′

u9 5 23308 1 3608 5 308

41.

x

y

=

θ =

π3

π

35

θ ′

u 55p

3, u9 5 2p 2

5p

35

p

3


x

y

=

θ =

π

6 π

65

−

θ ′

u9 57p

62 p 5

p

6

7p

6.u 5 2

5p

6

42.

x

y

=π4

θ =π

43

θ ′

u9 5 p 23p

45

p

4

44.

x

y

=π3

θ =π

32

−θ ′

u9 522p

31 p 5

p

3

45.

u9 5 2088 2 1808 5 288

= 208°

x

y

θ

θ

′ = 28°

u 5 2088 46.

u9 5 3608 2 3228 5 388

x

y

= 322°

θ

θ

′ = 38°

u 5 3228

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47.

u9 5 3608 2 2928 5 688

x

y

= −292°θ

θ′ = 68°

u 5 22928 48. lies in

Quadrant III.

Reference angle:

1808 2 1658 5 158x

y

−1658=

θ ′ 158=

θ

u 5 21658


x

y

π

511

π

5

=

θ

θ

′ =

u9 5p

5

p

5.u 5

11p

5

51. lies in Quadrant III.

Reference angle:

x

y

−1.8=θ ′ 1.342=

θ

p 2 1.8 < 1.342

u 5 21.8

50.

x

y

π

717

π

73

θ

θ

′ =

=

u9 517p

72

14p

75

3p

7

52. lies in Quadrant IV.

Reference angle:

x

y

θ ′= 1.358

θ = 4.5

4.5 2 p < 1.358

u

53. Quadrant III

tan 2258 5 tan 458 5 1

cos 2258 5 2cos 458 5 2!2

2

sin 2258 5 2sin 458 5 2!2

2

u9 5 458,

55. is coterminal with Quadrant IV.

tans27508d 5 2tan 308 5 2!3

3

coss27508d 5 cos 308 5!3

2

sins27508d 5 2sin 308 5 21

2

u9 5 3608 2 3308 5 308

3308,u 5 27508

54. Quadrant IV

sin

cos

tan 3008 5 2tan 608 5 2!3

3008 5 cos 608 51

2

3008 5 2sin 608 5 2!3

2

u 5 3008, u9 5 3608 2 3008 5 608,

56. Quadrant III

tans24958d 5 tan 458 5 1

coss24958d 5 2cos 458 5 2!2

2

sins24958d 5 2sin 458 5 2!2

2

u 5 24958, u9 5 458,

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57. Quadrant IV

tan15p

3 2 5 2tan1p

32 5 2!3

cos15p

3 2 5 cos1p

32 51

2

sin15p

3 2 5 2sin1p

32 5 2!3

2

u9 5 2p 25p

35

p

3

u 55p

3, 58.

sin

cos

tan 3p

45 21

3p

45 2

!2

2

3p

45

!2

2

u 53p

4, u9 5 p 2

3p

45

p

4

59. Quadrant IV

tan12p

62 5 2tan p

65 2

!3

3

cos12p

62 5 cos p

65

!3

2

sin12p

62 5 2sin p

65 2

1

2

u9 5p

6,

61. Quadrant II

tan 11p

45 2tan

p

45 21

cos 11p

45 2cos

p

45 2

!2

2

sin 11p

45 sin

p

45

!2

2

u9 5p

4,

60. ,

sin

cos

tan124p

3 2 5 2!3

124p

3 2 521

2

124p

3 2 5!3

2

u9 5p

3u 5

24p

3


in Quadrant III.

sin

cos

tan 10p

35 tan

p

35 !3

10p

35 2cos

p

35 2

1

2

10p

35 2sin

p

35 2

!3

2

u9 54p

32 p 5

p

3

4p

3.u 5

10p

3


Quadrant III

tan1217p

6 2 5 tan1p

62 5!3

3

cos1217p

6 2 5 2cos1p

62 5 2!3

2

sin1217p

6 2 5 2sin1p

62 5 21

2

u9 57p

62 p 5

p

6,

7p

6.u 5 2

17p

664.

sin

cos

tan1220p

3 2 5 !3

1220p

3 2 521

2

1220p

3 2 52!3

2

u 5220p

3, u9 5

p

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65.

in Quadrant IV.

cos u 54

5

cos u > 0

cos2 u 516

25

cos2 u 5 1 29

25

cos2 u 5 1 2 123

522

cos2 u 5 1 2 sin2 u

sin2 u 1 cos2 u 5 1

sin u 5 23

566.

sin u 51

csc u5

1

!105

!10

10

csc u 51

sin u

!10 5 csc u

csc u > 0 in Quadrant II.

10 5 csc2 u

1 1 s23d2 5 csc2 u

1 1 cot2 u 5 csc2 u

cot u 5 23

67.

cot u 5 2!3

cot u < 0 in Quadrant IV.

cot2 u 5 3

cot2 u 5 s22d2 2 1

cot2 u 5 csc2 u 2 1

1 1 cot2 u 5 csc2 u

csc u 5 22

69.

tan u 5!65

4

tan u > 0 in Quadrant III.

tan2 u 565

16

tan2 u 5 129

422

2 1

tan2 u 5 sec2 u 2 1

1 1 tan2 u 5 sec2 u

sec u 5 29

4

68.

sec u 51

5y85

8

5

cos u 51

sec u ⇒ sec u 5

1

cos u

cos u 55

8

70.

Quadrant IV ⇒ csc u 5 2!41

5

csc u 5 ±!41

5

1 116

255 csc2 u

1 1 cot2 u 5 csc2 u

tan u 5 25

4 ⇒ cot u 5 2

4

5

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71. is in Quadrant II.

cot u 51

tan u5

2!21

2

sec u 51

cos u5

25

!215

25!21

21

csc u 51

sin u5

5

2

tan u 5sin u

cos u5

2y5

2!21y55

22

!215

22!21

21

cos u 5 2!1 2 sin2 u 5 2!1 24

255 2

!21

5

sin u 52

5 and cos u < 0 ⇒ u

72. is in Quadrant III.

sec u 51

cos u5 2

7

3

csc u 51

sin u5

27

2!105

27!10

20

cot u 51

tan u5

3

2!105

3!10

20

tan u 5sin u

cos u5

22!10y7

23y75

2!10

3

sin u 5 2!1 2 cos2 u 5 2!1 29

495

2!40

75

22!10

7

cos u 5 23

7 and sin u < 0 ⇒ u

73. is in Quadrant II.

cot u 51

tan u5 2

1

4

csc u 51

sin u5

17

4!175

!17

4

sin u 5 tan u cos u 5 s24d12!17

17 2 54!17

17

cos u 51

sec u5

21

!175

2!17

17

sec u 5 2!1 1 tan2 u 5 2!1 1 16 5 2!17

tan u 5 24 and cos u < 0 ⇒ u 74. is in Quadrant II.

csc u 51

sin u5

26

!265 !26

sin u 5 tan u cos u 5 121

52125!26

26 2 5!26

26

cos u 51

sec u5

25

!265

25!26

26

sec u 5 2!1 1 tan2 u 5 2!1 11

255

2!26

5

tan u 51

cot u5

21

5

cot u 5 25 and sin u > 0 ⇒ u

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78. sec 2358 51

cos 2358< 21.7434

75. is in Quadrant IV.

cot u 51

tan u5

2!5

2

tan u 5sin u

cos u5

22y3

!5y35

22

!55

22!5

5

sec u 51

cos u5

3

!55

3!5

5

cos u 5 !1 2 sin2 u 5!1 24

95

!5

3

sin u 51

csc u5 2

2

3

csc u 5 23

2 and tan u < 0 ⇒ u 76. is in Quadrant III.

cot u 51

tan u5

3

!75

3!7

7

tan u 5sin u

cos u5

2!7y4

23y45

!7

3

csc u 51

sin u5 2

4

!75

24!7

7

sin u 5 2!1 2 cos2 u 5 2!1 29

165 2

!7

4

cos u 51

sec u5 2

3

4

sec u 5 24

3 and cot u > 0 ⇒ u

77. sin 108 < 0.1736 79. tan 2458 < 2.1445

81. coss21108d < 20.342080. csc 3208 51

sin 32085 21.5557 82.

< 21.1918

cots22208d 51

tans22208d

84. csc 0.33 51

sin 0.33< 3.0860

86. tan 11p

9< 0.8391 88. cos1215p

14 2 < 20.9749

83.

< 5.7588

secs22808d 51

coss22808d 85. tan12p

9 2 < 0.8391

87.

< 22.9238

csc128p

9 2 51

sin128p

9 2

89. (a) reference angle is or

and is in Quadrant I or Quadrant II.

Values in degrees:

Values in radian:

(b) reference angle is or

and is in Quadrant III or Quadrant IV.

Values in degrees:

Values in radians:7p

6,

11p

6

2108, 3308

up

6

308sin u 5 21

2 ⇒

p

6,

5p

6

308, 1508

up

6

308sin u 51

2 ⇒ 90. (a) cos reference angle is 45º or

and is in Quadrant I or IV.

Values in degrees:

Values in radians:

(b) cos reference angle is or

and is in Quadrant II or III.

Values in degrees:


4,

5p

4

1358, 2258

up

4

458u 5 2!2

2 ⇒

p

4,

7p

4

458, 3158

up

4

u 5!2

2 ⇒

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92. (a) csc

Reference angle is

Values in degrees:

Values in radians:

(b) csc

Reference angle is

Values in degrees:

Values in radians:p

6,

5p

6

308, 1508

p

6 or 308.

sin u 51

2u 5 2 ⇒

5p

4,

7p

4

2258, 3158

458 or p

4.

sin u 521

!2u 5 2!2 ⇒ 91. (a) reference angle is or

and is in Quadrant I or Quadrant II.

Values in degrees:

Values in radians:


and is in Quadrant II or Quadrant IV.

Values in degrees:


4,

7p

4

1358, 3158

up

4

458cot u 5 21 ⇒

p

3,

2p

3

608, 1208

up

3

608csc u 52!3

3 ⇒

93. (a) reference angle is or

and is in Quadrant II or Quadrant III.

Values in degrees:

Values in radians:


and is in Quadrant II or Quadrant III.

Values in degrees:


3,

4p

3

1208, 2408

u

608,p

3cos u 5 2

1

2 ⇒

5p

6,

7p

6

1508, 2108

u308,

p

6sec u 5 2

2!3

3 ⇒ 94. (a)

Reference angle is

Values in degrees:

Values in radians:

(b) Values in degrees:

Values in radians:p

4 or

7p

4

458 or 3158

5p

6,

11p

6

1508, 3308

p

6 or 308.

cot u 5 2!3 ⇒ cos u

sin u5 2!3

95. (a)

(b)

(c) fcos 308g2 5 1!3

2 22

53

4

cos 308 2 sin 308 5!3 2 1

2

fsud 1 gsud 5 sin 308 1 cos 308 51

21

!3

25

1 1 !3

2

96. (a)

(b)

(c) fcos 608g2 5 11

222

51

4

cos 608 2 sin 608 51

22

!3

25

1 2 !3

2

fsud 1 gsud 5 sin 608 1 cos 608 5!3

21

1

25

!3 1 1

2

(d)

(e)

(f) coss2308d 5 cos 308 5!3

2

2 sin 308 5 211

22 5 1

sin 308 cos 308 5 11

221!3

2 2 5!3

4

(d)

(e)

(f) coss2608d 5 cos 608 51

2

2 sin 608 5 2!3

25 !3

sin 608 cos 608 5 1!3

2 211

22 5!3

4

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97. (a)

(b)

(c) fcos 3158g2 5 1!2

2 22

51

2

cos 3158 2 sin 3158 5!2

22 12!2

2 2 5 !2

fsud 1 gsud 5 sin 3158 1 cos 3158 5 2!2

21

!2

25 0

98. (a)

(b)

(c)

(d) sin 2258 cos 2258 5 12!2

2 212!2

2 2 51

2

fcos 2258g2 5 12!2

2 22

51

2

cos 2258 2 sin 2258 52!2

22 12!2

2 2 5 0

fsud 1 gsud 5 sin 2258 1 cos 2258 5 2!2

22

!2

25 2!2

(d)

(e)

(f) coss23158d 5 coss3158d 5!2

2

2 sin 3158 5 212!2

2 2 5 2!2

sin 3158 cos 3158 5 12!2

2 21!2

2 2 5 21

2

(e)

(f) coss22258d 5 coss2258d 5 2!2

2

2 sin 2258 5 212!2

2 2 5 2!2

99. (a)

(b)

(c)

(d) sin 1508 cos 1508 51

2?

2!3

25

2!3

4

fcos 1508g2 5 12!3

2 22

53

4

cos 1508 2 sin 1508 52!3

22

1

25

21 2 !3

2

fsud 1 gsud 5 sin 1508 1 cos 1508 51

21

2!3

25

1 2 !3

2

(e)

(f ) coss21508d 5 coss1508d 52!3

2

2 sin 1508 5 211

22 5 1

100. (a)

(b)

(c)

(d) sin 3008 cos 3008 5 12!3

2 211

22 52!3

4

fcos 3008g2 5 11

222

51

4

cos 3008 2 sin 3008 51

22 12!3

2 2 51 1 !3

2

fsud 1 gsud 5 sin 3008 1 cos 3008 52!3

21

1

25

1 2 !3

2

(e)

(f) coss23008d 5 coss3008d 51

2

2 sin 3008 5 212!3

2 2 5 2!3

101. (a)

(b)

(c)

(d) sin 7p

6 cos

7p

65 12

1

2212!3

2 2 5!3

4

3cos 7p

6 42

5 12!3

2 22

53

4

cos 7p

62 sin

7p

65

2!3

22 12

1

22 51 2 !3

2

fsud 1 gsud 5 sin 7p

61 cos

7p

65 2

1

22

!3

25

21 2 !3

2

(e)

(f) cos127p

6 2 5 cos17p

6 2 52!3

2

2 sin 7p

65 212

1

22 5 21

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102. (a)

(b)

(c)

(d) sin 5p

6 cos

5p

65 11

2212!3

2 2 52!3

4

3cos 5p

6 42

5 12!3

2 22

53

4

cos 5p

62 sin

5p

65

2!3

22

1

25

21 2 !3

2


61 cos

5p

65

1

21

2!3

25

1 2 !3

2

(e)

(f) cos125p

6 2 5 cos15p

6 2 52!3

2

2 sin 5p

65 211

22 5 1

103. (a)

(b)

(c)

(d) sin 4p

3 cos

4p

35 12!3

2 2121

22 5!3

4

3cos 4p

3 42

5 121

222

51

4

cos 4p

32 sin

4p

35 2

1

22 12!3

2 2 5!3 2 1

2


31 cos

4p

35

2!3

22

1

25

21 2 !3

2

(e)

(f) cos124p

3 2 5 cos14p

3 2 5 21

2

2 sin 4p

35 212!3

2 2 5 2!3

104. (a)

(b)

(c)

(d) sin 5p

3 cos

5p

35 12!3

2 211

22 52!3

4

3cos 5p

3 42

5 11

222

51

4

cos 5p

32 sin

5p

35

1

22 12!3

2 2 51 1 !3

2


31 cos

5p

35

2!3

21

1

25

1 2 !3

2

(e)

(f) cos125p

3 2 5 cos15p

3 2 51

2

2 sin 5p

35 212!3

2 2 5 2!3

105. (a)

(b)

(c) fcos 2708g25 02

5 0

cos 2708 2 sin 2708 5 0 2 s21d 5 1

fsud 1 gsud 5 sin 2708 1 cos 2708 5 21 1 0 5 21 (d)

(e)

(f) coss22708d 5 coss2708d 5 0

2 sin 2708 5 2s21d 5 22

sin 2708 cos 2708 5 s21ds0d 5 0

106. (a)

(b)

(c) fcos 1808g25 s21d2

5 1

cos 1808 2 sin 1808 5 21 2 0 5 21

fsud 1 gsud 5 sin 1808 1 cos 1808 5 0 2 1 5 21 (d)

(e)

(f) coss21808d 5 coss1808d 5 21

2 sin 1808 5 2s0d 5 0

sin 1808 cos 1808 5 0s21d 5 0

107. (a)

(b)

(c) 3cos 7p

2 42

5 025 0

cos 7p

22 sin

7p

25 0 2 s21d 5 1


21 cos

7p

25 21 1 0 5 21 (d)

(e)

(f) cos127p

2 2 5 cos17p

2 2 5 0

2 sin 7p

25 2s21d 5 22

sin 7p

2 cos

7p

25 s21ds0d 5 0

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108. (a)

(b)

(c) 3cos 5p

2 42

5 025 0

cos 5p

22 sin

5p

25 0 2 1 5 21


21 cos

5p

25 1 1 0 5 1 (d)

(e)

(f) cos125p

2 2 5 cos15p

2 2 5 0

2 sin 5p

25 2s1d 5 2

sin 5p

2 cos

5p

2 5 s1ds0d 5 0

109.

(a) January:

(b) July:

(c) December: t 5 12 ⇒ T < 31.758

t 5 7 ⇒ T 5 708

t 5 1 ⇒ T 5 49.5 1 20.5 cos1p s1d6

27p

6 2 5 298

T 5 49.5 1 20.5 cos1p t

62

7p

6 2

110.

(a) thousand

(b) thousand

(c) thousand

(d) thousand

Answers will vary.

Ss6d < 25.8

Ss5d < 27.5

Ss14d < 33.0

Ss1d 5 23.1 1 0.442s1d 1 4.3 sin1p

62 < 25.7

S 5 23.1 1 0.442t 1 4.3 sin1pt

6 2, t 5 1 ↔ Jan. 2004

111. sin

(a)

miles

(b)

miles

(c)

milesd 56

sin 1208< 6.9

u 5 1208

d 56

sin 9085

6

15 6

u 5 908

d 56

sin 3085

6

s1y2d5 12

u 5 308

u 56

d ⇒ d 5

6

sin u

113. True. The reference angle for is

and sine is positive

in Quadrants I and II.

u9 5 1808 2 1518 5 298,

u 5 1518

112. As increases from to x decreases from

12 cm to 0 cm and y increases from 0 cm to

12 cm. Therefore, increases from 0

to 1, and decreases from 1 to 0.

Thus, begins at 0 and increases

without bound. When the tangent

is undefined.

u 5 908,

tan u 5 yyx

cos u 5 xy12

sin u 5 yy12

908,08u

114. False. and

cot12p

42 5 21

2cot13p

4 2 5 2s21d 5 1

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115. (a)

(b) It appears that sin u 5 sins1808 2 ud.

0 0.3420 0.6428 0.8660 0.9848

0 0.3420 0.6428 0.8660 0.9848sins1808 2 ud

sin u

80860840820808u

116.

Patterns and conclusions may vary.

Function

Domain All reals except

Range

Evenness No Yes No

Oddness Yes No Yes

Period

Zeros npp

21 npnp

p2p2p

s2`, `df21, 1gf21, 1g

p

21 nps2`, `ds2`, `d

tan xcos xsin x

Function

Domain All reals except All reals except All reals except

Range

Evenness No Yes No

Oddness Yes No Yes

Period

Zeros None Nonep

21 np

p2p2p

s2`, `ds2`, 21g < f1, `ds2`, 21g < f1, `d

npp

21 npnp

cot xsec xcsc x

118.

x 5 2179 < 21.889

9x 5 217

44 2 9x 5 61

120.

x < 1.186, 21.686

x 521 ±!1 1 4s4ds2d

45

21 ±!33

4

2x2 1 x 2 4 5 0

117.

x 5 7

3x 5 21

3x 2 7 5 14

119.

x < 3.449, x < 21.449

x 52 ±!4 1 20

25 1 ±!6

x2 2 2x 2 5 5 0

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Section 4.5 Graphs of Sine and Cosine Functions

Section 4.5 Graphs of Sine and Cosine Functions 317

121.

x < 25.908, 4.908

x 521 ±!1 1 4s29d

25

21 ±!117

2

x2 1 x 2 29 5 0

27 5 sx 2 1dsx 1 2d

3

x 2 15

x 1 2

9

123.

x 5 3 2 log4 726 5 3 2ln 726

ln 4< 21.752

3 2 x 5 log4 726

432x 5 726

125.

x 5 e26 < 0.002479 < 0.002

ln x 5 26

122.

( extraneous)x 5 0x 5 6

xsx 2 6d 5 0

x2 2 6x 5 0

10x 5 x2 1 4x

5

x5

x 1 4

2x

124.

x 51

2 ln 86 < 2.227

2x 5 ln 86

86 5 e2x

90 5 4 1 e2x

4500

4 1 e2x5 50

126. ⇒ x 1 10 5 e2 ⇒ x 5 e2 2 10 < 22.611ln !x 1 10 512 lnsx 1 10d 5 1 ⇒ lnsx 1 10d 5 2

n You should be able to graph and

n Amplitude:

n Period:

n Shift: Solve and

n Key increments: (period)1

4

bx 2 c 5 2p.bx 2 c 5 0

2p

|b|

|a|y 5 a cossbx 2 cd.y 5 a sinsbx 2 cd

Vocabulary Check

1. amplitude 2. one cycle 3. 4. phase shift2p

b

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1.

(a) -intercepts:

(b) -intercept:

(c) Increasing on:

Decreasing on:

(d) Relative maxima:

Relative minima: 12p

2, 212, 13p

2, 212

123p

2, 12, 1p

2, 12

123p

2, 2

p

22, 1p

2,

3p

2 2

122p, 23p

2 2, 12p

2,

p

22, 13p

2, 2p2

s0, 0dy

s22p, 0d, s2p, 0d, s0, 0d, sp, 0d, s2p, 0dx

fsxd 5 sin x

2.

(a) -intercepts:

(b) -intercept:

(c) Increasing on:

Decreasing on:

(d) Relative maxima:

Relative minima: s2p, 21d, sp, 21d

s22p, 1d, s0, 1d, s2p, 1d

s22p, 2pd, s0, pd

s2p, 0d, sp, 2pd

s0, 1dy

123p

2, 02, 12

p

2, 02, 1p

2, 02, 13p

2, 02x

fsxd 5 cos x

7.

Period:

Amplitude: |23| 52

3

2p

p5 2

y 52

3 sin px

3.

Period:

Amplitude: |3| 5 3

2p

25 p

y 5 3 sin 2x

5.

Period:

Amplitude: |52| 55

2

2p

1y25 4p

y 55

2 cos

x

2

Xmin = -2

Xmax = 2

Xscl =

Ymin = -4

Ymax = 4

Yscl = 1

py2

p

p

Xmin = -4

Xmax = 4

Xscl =

Ymin = -3

Ymax = 3

Yscl = 1

p

p

p

4.

Period:

Amplitude: |a| 5 2

2p

b5

2p

3

y 5 2 cos 3x

6.

Period:

Amplitude: |a| 5 |23| 5 3

2p

b5

2p

s1y3d5 6p

y 5 23 sin x

3

8.

Period:

Amplitude: |a| 53

2

2p

b5

2p

spy2d5 4

y 53

2 cos

px

2

Xmin = -

Xmax =

Xscl =

Ymin = -3

Ymax = 3

Yscl = 1

py4

p

p

Xmin = -

Xmax =

Xscl =

Ymin = -4

Ymax = 4

Yscl = 1

p

6p

6p

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9.

Period:

Amplitude: |22| 5 2

2p

15 2p

y 5 22 sin x 10.

Period:

Amplitude: |a| 5 |21| 5 1

2p

b5

2p

2y55 5p

y 5 2cos 2x

511.

Period:

Amplitude: |14| 51

4

2p

2y35 3p

y 51

4 cos

2x

3

13.

Period:

Amplitude: |13| 51

3

2p

4p5

1

2

y 51

3 sin 4px

15.

The graph of is a horizontal shift to the right

units of the graph of (a phase shift).fp

g

gsxd 5 sinsx 2 pd

f sxd 5 sin x

17.

The graph of is a reflection in the axis of the

graph of f.

x-g

gsxd 5 2cos 2x

f sxd 5 cos 2x

19.

The graph of has five times the amplitude of

and reflected in the axis.x-

f,g

gsxd 5 25 cos x

fsxd 5 cos x

21.

The graph of is a vertical shift upward of five

units of the graph of f.

g

gsxd 5 5 1 sin 2x

f sxd 5 sin 2x

12.

Period:

Amplitude: |a| 55

2

2p

b5

2p

1y45 8p

y 55

2 cos

x

414.

Amplitude: |a| 53

4

Period: 2p

b5

2p

py125 24

y 53

4 cos

p x

12

16.

g is a horizontal shift of f units to the left.p

fsxd 5 cos x, gsxd 5 cossx 1 pd

18.

g is a reflection of f about the y-axis.

(or, about the -axis)x

f sxd 5 sin 3x, gsxd 5 sins23xd

20.

The amplitude of g is one-half that of f. g is a

reflection of f in the x-axis.

fsxd 5 sin x, gsxd 5 212 sin x

22.

g is a vertical shift of f six units downward.

f sxd 5 cos 4x, gsxd 5 26 1 cos 4x

24. The period of g is one-half the period of f.23. The graph of has twice the amplitude as the graph

of The period is the same.f.

g

25. The graph of is a horizontal shift units to the

right of the graph of f.

pg 26. Shift the graph of f two units upward to obtain the

graph of g.

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27.

Period:

Amplitude: 1

Period:

Amplitude:

−2

−3

−4

1

4

x

y

π

2π

23

−

f

g

|24| 5 4

2p

gsxd 5 24 sin x

2p

f sxd 5 sin x

29.

Period:

Amplitude: 1

is a vertical shift of the graph of

four units upward.

f

g

ππ−−1

−2

22

2

3

4

6

x

y

fsxdgsxd 5 4 1 cos x

2p

fsxd 5 cos x

28.

–2

2

x

π6

fg

y

gsxd 5 sin x

3

fsxd 5 sin x

30.

–2

2

xπ

f

g

y

gsxd 5 2cos 4x

fsxd 5 2 cos 2x

x 0

0 1 0 0

0 1!3

2

!3

2

1

2sin

x

3

21sin x

2p3p

2p

p

2

x 0

2 0 0 2

1 1 2121212cos 4x

222 cos 2x

p3p

4

p

2

p

4

31.

Period:

Amplitude:

is the graph of

shifted vertically three units upward.

fsxdgsxd 5 3 21

2 sin

x

2

1

2

4p

x

y

π3π2 π4ππ4 −−

f

g

−2

−1

−3

−4

1

2

3

4

fsxd 5 21

2 sin

x

2

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33.

Period:

Amplitude: 2

is the graph of shifted

units to the left.

–3

3

xπ π2

f

g

y

p

fsxdgsxd 5 2 cossx 1 pd

2p

fsxd 5 2 cos x

36. fsxd 5 sin x, gsxd 5 2cos1x 1p

22

x 0

0 1 0 0

0 1 0 0212cos1x 1p

22

21sin x

2p3p

2p

p

2

32.

x

y

f

g−1

−2

−2 2−1 1

2

3

4

1

gsxd 5 4 sin px 2 2

fsxd 5 4 sin px

34.

f g

ππ

ππ

−

−

1

−1

22

x

y

gsxd 5 2cos1x 2p

22fsxd 5 2cos x

x 0 1 2

0 4 0 0

2 22262222gsxd

24f sxd

3

2

1

2

x 0

0 1 0

0 0 1 0212cos1x 2p

2221212cos x

2p3p

2p

p

2

35.

Period:

Amplitude: 1

−2

−2

2

2

f = g

pp

2p

sin x 5 cos1x 2p

22

f sxd 5 sin x, gsxd 5 cos1x 2p

22

Conjecture: sin x 5 2cos1x 1p

22

−2

−2

2

2

pp

f = g

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38.

Conjecture: cos x 5 2cossx 2 pd

−2

−2

2

2

pp

f = g

fsxd 5 cos x, gsxd 5 2cossx 2 pd37.

Thus,

−2

−2

2

2

f = g

pp

f sxd 5 gsxd.

gsxd 5 2sin1x 2p

22 5 sin1p

22 x2 5 cos x

fsxd 5 cos x

x 0

1 0 0 1

1 0 0 1212cossx 2 pd

21cos x

2p3p

2p

p

2

39.

Period:

Amplitude: 3

Key points:

x

y

π

2π

2π

23 π

23

− −

−4

1

2

3

4

s2p, 0d13p

2, 232,sp, 0d,1p

2, 32,s0, 0d,

2p

y 5 3 sin x

41.

Period:

Amplitude: 1

Key points:

x

y

−1

−2

−3

2

3

π2 π4

s2p, 21d, s3p, 0d, s4p, 1ds0, 1d, sp, 0d,

4p

y 5 cos x

2

40.

Period:

Amplitude:

x

y

−0.5

−1

0.5

1

π π

2π

2π2

−

1

4

2p

y 51

4 cos x

42.

Period:

Amplitude:

x

y

π

8π

8

−2

1

2

3π

83 π

85

−

1

2p

45

p

2

y 5 sin 4x

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43.

Period:

Amplitude: 1

Shift: Set and

Key points: 1p

4, 02, 13p

4, 12, 15p

4, 02, 17p

4, 212, 19p

4, 02

x 59p

4x 5

p

4

x 2p

45 2px 2

p

45 0

2p

–3

–2

1

2

3

xπ−π

yy 5 sin1x 2p

42; a 5 1, b 5 1, c 5p

4

44.

Horizontal shift units to the rightp

x

y

π

2π

2−

−1

−2

1

2

π

23

y 5 sinsx 2 pd

45.

Period:

Amplitude: 8

Key points: s2p, 28d, 12p

2, 02, s0, 8d, 1p

2, 02, sp, 28d

2p

x

y

−2

−4

−6

−8

−10

2

8

10

πππ2 π2− −

y 5 28 cossx 1 pd

46.

Amplitude: 3

Horizontal shift units to the left

Note: 3 cos1x 1p

22 5 23 sin x

1

3

4

−2

−3

−4

x

y

π

2

π

23π

2−

p

2

y 5 3 cos1x 1p

22 47.

Amplitude: 2

Period:

6−6

−4

4

2p

2py35 3

y 5 22 sin 2px

3

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48.

Amplitude: 10

Period:

18−18

−12

12

2p

py65 12

y 5 210 cos px

649.

Amplitude: 5

Period:

30−30

−20

20

2p

py125 24

y 5 24 1 5 cos p t

1250.

Amplitude: 2

Period:

6−6

−2

6

2p

2py35 3

y 5 2 2 2 sin 2px

3

51.

Amplitude:

Period:

−2

2

4π 5π−

2p

1y25 4p

2

3

y 52

3 cos1x

22

p

42 52.

Amplitude: 3

Period:

−6

6

π

2

π

2−

2p

65

p

3

y 5 23 coss6x 1 pd 53.

Amplitude: 2

Period:

−

−4

4

pp

p

2

y 5 22 sins4x 1 pd

55.

Amplitude: 1

Period: 1

−3

−1

3

3

y 5 cos12px 2p

22 1 154.

Amplitude: 4

Period:

−8

8

12−12

3p

y 5 24 sin12

3x 2

p

32 56.

Amplitude: 3

Period: 4

−7

1

6−6

y 5 3 cos1px

21

p

22 2 3

58.

Amplitude: 5

Period:

−

0

12

pp

p

y 5 5 cossp 2 2xd 1 657.

Amplitude: 5

Period:

−

−4

20

pp

p

y 5 5 sinsp 2 2xd 1 10 59.

Amplitude:

Period:

−

−0.02

0.02

p

180

p

180

160

1100

y 51

100 sin 120p t©

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60.

Amplitude:

Period:

−0.04

0.04

0.06−0.06

1

25

1

100

y 521

100 coss50p td 61.

Amplitude:

Since is the graph of

reflected about the

axis and shifted vertically four

units upward, we have

and Thus,

5 4 2 4 cos x.

fsxd 5 24 cos x 1 4

d 5 4.

a 5 24

x-

gsxd 5 4 cos x

f sxd

1

2f8 2 0g 5 4

fsxd 5 a cos x 1 d

63.

Amplitude:

Graph of is the graph of

reflected about the

-axis and shifted vertically one

unit upward. Thus,

f sxd 5 26 cos x 1 1.

x

gsxd 5 6 cos x

f

1

2f7 2 s25dg 5 6

f sxd 5 a cos x 1 d 65.

Amplitude:

Since the graph is reflected

about the axis, we have

Period:

Phase shift:

Thus, f sxd 5 23 sin 2x.

c 5 0

2p

b5 p ⇒ b 5 2

a 5 23.

x-

|a| 5 3

f sxd 5 a sinsbx 2 cd

67.

Amplitude:

Period:

Phase shift: when

Thus, f sxd 5 sin1x 2p

42.

s1d1p

42 2 c 5 0 ⇒ c 5p

4

x 5p

4.bx 2 c 5 0

2p ⇒ b 5 1

a 5 1

f sxd 5 a sinsbx 2 cd

62.

Amplitude:

Reflected in the x-axis:

y 5 23 2 cos x

d 5 23

24 5 21 cos 0 1 d

a 5 21

22 2 s24d2

5 1

fsxd 5 a cos x 1 d

64.

Reflected in -axis,

y 5 24 21

2 cos x

d 5 24

a 5 21

2x

Period: 2p

Amplitude: 1

2

y 5 a cos x 1 d

66.

Amplitude:

Period:

Phase shift:

y 5 2 sin1x

22c 5 0

2p

b5 4p ⇒ b 5

1

2

4p

2 ⇒ a 5 2

y 5 a sinsbx 2 cd

68.

Amplitude:

Period:

Phase shift:

y 5 2 sin1px

21

p

22

c

b5 21 ⇒ c 5 2

p

2

2p

b5 4 ⇒ b 5

p

2

4

2 ⇒ a 5 2

y 5 a sinsbx 2 cd 69.

In the interval when

−2

−2

2

2

pp

y1

y2

x 5 25p

6, 2

p

6,

7p

6,

11p

6.

sin x 5 212f22p, 2pg,

y2 5 21

2

y1 5 sin x

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70.

.

−2

−2

2

2

pp

y1

y2

y1 5 y2 when x 5 p, 2p

y2 5 21

y1 5 cos x

72.

(a)

(b) Maximum sales: December

Minimum sales: June st 5 6d

st 5 12d

1

0

12

120

S 5 74.50 1 43.75 cos p t

6

71.

(a)

(b)

(c) Cycles per min cycles per min

(d) The period would change.

560

65 10

Time for one cycle 5 one period 52p

py35 6 sec

0

−2

4

2

p

v 5 0.85 sin pt

3

73.

(a)

(b) Minimum:

Maximum: 30 1 25 5 55 feet

30 2 25 5 5 feet

0

0

135

60

h 5 25 sin p

15st 2 75d 1 30

74.

1 heartbeat

s3y4d ⇒ 4

3 heartbeatsysecond 5 80 heartbeatsyminute

Period: 2p

s8py3d 53

4

0

60

1.5

130

P 5 100 2 20 cos 8p

3t

75.

(a)

This is to be expected:

(b) The constant 30.3 gallons is the average daily

fuel consumption.

365 days 5 1 year

Period: 2p

b5

2p

s2py365d5 365 days

C 5 30.3 1 21.6 sin12pt

3651 10.92

(c)

Consumption exceeds 40 when

(Graph together with

(Beginning of May through part of September)

y 5 40.)C124 ≤ x ≤ 252.

gallonsyday

0

0

365

60

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76. (a) Yes, y is a function of t because for each

value of t there corresponds one and only

one value of y.

(b) The period is approximately

The amplitude is approximately12s2.35 2 1.65d 5 0.35 centimeters.

2s0.375 2 0.125d 5 0.5 seconds.

(c) One model is

(d)

0

0

0.9

4

y 5 0.35 sin 4p t 1 2.

77. (a)

(b) y 5 0.506 sins0.209x 2 1.336d 1 0.526

−10 10

1.0

2.0

0.5

1.5

20 30 40 50 60 70

−0.5

x

y (c)

The model is a good fit.

(d) The period is

(e) June 29, 2007 is day 545. Using the model,

or 27.09%.y < 0.2709

2p

0.209< 30.06.

−10 10

1.0

2.0

1.5

20 30 40 50 60 70

−0.5

x

y

78. (a)

(b)

The model somewhat fits the data.

(c)

The model is a good fit.

0

45

12

95

0

0

12

80

Astd 5 19.73 sins0.472t 2 1.74d 1 70.2 (d) Nantucket:

Athens:

The constant term (d) gives the average daily high

temperature.

(e) Period for

Period for

You would expect the period to be 12 (1 year).

(f ) Athens has greater variability. This is given by the

amplitude.

Astd 52p

0.472< 13

Nstd 52p

s2py11d 5 11

70.28

588

79. True. The period is 2p

3y105

20p

3.

81. True

80. False. The amplitude is that of y 5 cos x.1

2

82. Answers will vary.

83. The graph passes through and has period

Matches (e).

p.s0, 0d 84. The amplitude is 4 and the period Since

is on the graph, matches (a).s0, 24d2p.

85. The period is and the amplitude is 1. Since

and are on the graph, matches (c).sp, 0ds0, 1d4p 86. The period is Since is on the graph,

matches (d).

s0, 21dp.

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87. (a) is even.

−

−2

2

pp

h(x) = cos2 x

hsxd 5 cos2 x (b) is even.

−

−2

2

pp

h(x) = sin2 x

hsxd 5 sin2 x (c) is odd.

−

−2

2

pp

h(x) = sin x cos x

hsxd 5 sin x cos x

88. (a) In Exercise 87, is even and we saw that is even.

Therefore, for even and we make the conjecture that is even.

(b) In Exercise 87, is odd and we saw that is even.

Therefore, for odd and we make the conjecture that is even.

(c) From part (c) of 87, we conjecture that the product of an even function and an odd

function is odd.

hsxdhsxd 5 fgsxdg2,gsxdhsxd 5 sin2xgsxd 5 sin x

hsxdhsxd 5 f fsxdg2,f sxdhsxd 5 cos2 xfsxd 5 cos x

89. (a)

0.8415 0.9983 1.0 1.0sin x

x

20.00120.0120.121x

0 0.001 0.01 0.1 1

Undef. 1.0 1.0 0.9983 0.8415sin x

x

x

(b)

As approaches 1.

(c) As approaches 0, approaches 1.sin x

xx

x → 0, f sxd 5sin x

x

−1 1

0.8

1.1

90. (a) (b)

As approaches 0.

(c) As approaches 0, approaches 0.1 2 cos x

xx

x → 0, 1 2 cos x

x

1−1

0.5

−0.5

20.000520.00520.0520.45971 2 cos x

x

20.00120.0120.121x

0 0.001 0.01 0.1 1

Undef. 0.0005 0.005 0.05 0.45971 2 cos x

x

x

91. (a)

(b)

−2π π

−2

2

2

−2π π

−2

2

2

(c) Next term for sine approximation:

Next term for cosine approximation:

−2π π

−2

2

2−2π π

−2

2

2

2x6

6!

2x7

7!

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Section 4.6 Graphs of Other Trigonometric Functions

Section 4.6 Graphs of Other Trigonometric Functions 329

92. (a) (d)

(b) (e)

(c) (f )

In all cases, the approximations are very accurate.

cos121

22 < 0.8776sin1p

82 < 0.3827

cos12p

42 < 0.7071sin11

22 < 0.4794

coss21d < 0.5403sins1d < 0.8415 93.

Slope 57 2 1

2 2 05 3

x

y

−1−2−3−4 1 2 3 4−1

7

6

5

4

3

2

1

(2, 7)

(0, 1)

95. 8.5 5 8.511808

p 2 < 487.014894.

m 522 2 4

3 1 15

26

45

23

2

−2

−1−1 1 2 3 4−2−3−4

−3

−4

1

2

3

4

x

y

(−1, 4)

(3, −2)

96. 20.48 5 20.4811808

p 2 < 227.5028 97. Answers will vary. (Make a Decision)

n You should be able to graph:

n When graphing or you should know to first graph

or since

(a) The intercepts of sine and cosine are vertical asymptotes of cosecant and secant.

(b) The maximums of sine and cosine are local minimums of cosecant and secant.

(c) The minimums of sine and cosine are local maximums of cosecant and secant.

n You should be able to graph using a damping factor.

y 5 a sinsbx 2 cdy 5 a cossbx 2 cdy 5 a cscsbx 2 cdy 5 a secsbx 2 cd

y 5 a cscsbx 2 cdy 5 a secsbx 2 cd

y 5 a cotsbx 2 cdy 5 a tansbx 2 cd

Vocabulary Check

1. vertical 2. reciprocal 3. damping

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C H A P T E R 4 Trigonometric Functions - Fraser HS Math

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