lial trig chapter 4 10e - · pdf filesection 4.3 graphs of the tangent and cotangent functions...

Copyright © 2013 Pearson Education, Inc. 32

Chapter 4 Graphs of the Circular Functions

Section 4.1 Graphs of the Sine and Cosine Functions

1. G

2. A

3. E

4. D

5. B

6. H

7. F

8. C

9. D

10. B

11. C

12. A

13. 2

14. 3

15. 2

3

16. 3

4

17. 1

18. 1

19. 2

20. 3

Section 4.1 Graphs of the Sine and Cosine Functions 33

Copyright © 2013 Pearson Education, Inc.

21. 1

22. The graph of y = sin (−x) is the same as the

graph of y = −sin x because the sine function is an odd function.

23. 4 ;1

24. 3 ;1

25. 8

;13

26. 6 ;1

27. 2

;13

28. ;1

29. 8 ;2

30. ;3

31. 2

;23

32. ;5

34 Chapter 4 Graphs of the Circular Functions


33. 2;1

34. 2;1

35. 1; 2

36. 1;3

37. 1

4;2

38. 2

8;3

39. 2;

40. 2;

41. 2cos 2y x=

42. 2sin 2y x=-

43. 12

3cosy x=-

44. 12

3cosy x=

45. 3sin 4y x=

46. 3cos 4y x=-

47. (a) 80°; 50°

(b) 15

(c) about 35,000 yr.

(d) downward.

48. (a) 120 mm; 80 mm

(b) 20

(c) 75

49. 24 hours

50. 1.2

51. 6 P.M; 0.2 ft

Section 4.1 Graphs of the Sine and Cosine Functions 35


52. 7:19 P.M; 0 ft

53. 3:18 A.M; 2.4 feet

54. (a) 2 hr

(b) 1 yr.

55. (a) 1

5;60

(b) 60 cycles are completed.

(c) 5;1.545; 4.045; 4.045;1.545- -

(d)

56. (a) 1

3.8;20

(b) 20

(c) 3.074;1.174; 3.074; 3.074;1.174- - -

(d)

57. (a)

(b) Maximums: 1 5 9

, , , ...4 4 4

x =

Minimums: 3 7 11

, , , ...4 4 4

x =

(c) Answers will vary.

58. (a) 2( ) .04 .6 330 7.5sin 2C x x x x

(b) Answers will vary.

(c) ( ) ( )( )

( )

20.04 1970

0.6 1970 330

7.5sin 2 1970

C x x

x

xπ

= -

+ - +

é ù+ -ë û

59. (a) 31

(b) 38

(c) 57

(d) 58

(e) 37

(f) 16

60. (a) 21.998 watts per m

(b) 246.461 watts per m-

(c) 246.478 watts per m

(d) Answers may vary. A possible solution is N = 82.5. Other answers are possible. Since N represents a day number, which should be a natural number, we might interpret day 82.5 as noon on the 82nd day.

61. 4

1;240 or 3

62. 2

1;120 or 3

63. No, we can’t say that sin sinbx b x= . If b is

not zero, then the period of siny bx= is 2

bπ

,

and the amplitude is 1. The period of siny b x= is 2π , and the amplitude is b .

64. No, we can’t say that cos cosbx b x= . If b is

not zero, then the period of cosy bx= is 2

bπ

,

and the amplitude is 1. The period of cosy b x= is 2π , and the amplitude is b .



65.X ≈ −0.4161468, Y ≈ 0.90929743. X is cos 2, and Y is sin 2.

66. X = 2, Y ≈ 0.90929743; sin 2 ≈ 0.90929743

67. X = 2, Y ≈ −0.4161468; cos 2 ≈ −0.4161468

68. Answers will vary. The x-coordinate is cosθ and the y-coordinate is sin .θ

Section 4.2 Translations of the Graphs of the Sine and Cosine Functions

1. D

2. G

3. H

4. A

5. B

6. E

7. I

8. C

9. C

10. B

11. A

12. D

13. The graph of sin 1y x= + is the graph of

siny x= translated vertically 1 unit up, while

the graph of sin( 1)y x= + is the graph of

siny x= shifted horizontally 1 unit left.

14. sin 1y x= +

15. B

16. D

17. C

18. A

19. If the graph of y = cos x is translated 2π units

horizontally to the right, it will coincide with the graph of y = sin x.

20. If the graph of y = sin x is translated 2π units

horizontally to the left, it will coincide with the graph of y = cos x.

21. y = −1 + sin x

22. y = 2 + cos x

23. ( )3cosy x π= -

24. ( )6cosy x π= -

25. 2;2 ; none; π units to the left

26. 3;2 ; none;

2

π units to the left

27. 1

;4 ;4

none; π units to the left

28. 1

;4 ;2

none; 2π units to the left

29. 3; 4, none; 1

2 unit to the right

30. 1; 2 none; 1

3 unit to the right

31. 2

1; ;3

up 2units;

15

π unit to the right

32. 1

; ;2 down 1 unit;

3

2

π units to the right

Section 4.2 Translations of the Graphs of the Sine and Cosine Functions 37


33.

34.

35.

36.

37.

38.

39.

40.

41.

42.



43.

44.

45.

46.

47.

48.

49.

50.

51.

52.

53.

Section 4.2 Translations of the Graphs of the Sine and Cosine Functions 39


54.

55.

56.

57. (a) Yes (b) This line represents the average yearly

temperature in Seattle of 53.5°F. (This is also the actual average yearly temperature.)

(c) 12.5; 12; 4.5

(d) ( ) ( )12.5sin 4.5 53.56

f x xπé ùê ú= - +ê úë û

(e) The functions gives an excellent model for the data

(f)

From the sine regression we have

( )12.41sin 0.53 2.26 52.42y x» - +

58. (a) 73.5 F (b) See the graph in part (d)

(c) ( ) ( )19.5cos 7 73.56

f x xé ùê ú= - +ê úë û

(d) The function gives an excellent model for the data.

( ) ( )19.5cos 7 73.56


(e)

From the sine regression we have

( )19.68sin 0.53 2.20 72.90y x= - +



59.

60.

Chapter 4 Quiz (Section 4.1−4.2)

1. 4; 2; 3 units up; 4


2. 4;2

3. 1

;2

4. 2; 3

5. 2 ;2

6. ;1

7.

8. 2siny x=

9. cos 2y x=

10. siny x=-

11. 73 F

12. 60ºF; 84ºF

Section 4.3 Graphs of the Tangent and Cotangent Functions

1. C

2. A

3. B

4. D

5. F

6. E

7.

Section 4.3 Graphs of the Tangent and Cotangent Functions 41


8.

9.

10.

11.

12.

13.

14.

15.

16.

17.



18.

19.

20.

21.

22.

23.

24.

25.

26.

27.

28.

Section 4.3 Graphs of the Tangent and Cotangent Functions 43


29.

30.

31.

32.

33. y = −2 tan x 34. y = −2 cot x 35. y = cot 3x

36. y = tan 3x

37. 12

1 tany x= +

38. y = −2 + 2 cot x

39. True

40. False

41. False

42. True.

43. four

44. domain:

( )2 1 , where is any integer .2

x x n nπì üï ïï ï¹ +í ýï ïï ïî þ

range: ( ),-¥ ¥

45. ( ) sin( ) sintan tan

cos( ) cos

x xx xx x

- -- = = =-

-,



46. ( )( )( )

cos coscot cot

sin sin

x xx xx x

-- = = =-

- -,

{ }, where is any integer .x x n nπ¹

47. (a) 0 m (b) –2.9 m

(c) –12.3 m

(d) 12.3 m

(e) t = 0.25 leads to tan ,2

π which is

undefined.

48. Answers will vary.

49. π

50. 5

4

51. 5

,4

x nπ π= +

52. approximately 0.3217505544 53. ≈ 3.463343208

54. | 0.3217505544 ,

where is an integer

x x nn

πì ü= +ï ïï ïí ýï ïï ïî þ



Section 4.4 Graphs of the Secant and Cosecant Functions

1. B

2. C

3. D

4. A

5.

6.

7.

8.

9.

10.

11.

12.

13.

Section 4.4 Graphs of the Secant and Cosecant Functions 45


14.

15.

16.

17.

18.

For exercises 19−24, other answers are possible.

19. y = sec 4x

20. y = sec 2x

21. y = −2 + csc x

22. y = 1 + csc x

23. y = −1 − sec x

24. 12

1 cscy x= -

25. True

26. False

27. True

28. True

29. None

30. domain: , where is any integer4

nx x nì üï ïï ï¹í ýï ïï ïî þ

range: ( , – 2] [2, )-¥ È ¥ since a = –2.

31. 1 1

sec( ) sec( )cos( ) cos( )

x xx x

- = = =-

,



32. ( )( )1 1

csc cscsin sin

x xx x

- = = =-- -

,

{ }, where is any integerx x n nπ¹

33. (a) 4 m

(b) 6.3 m

(c) 63.7 m

34. Answers will vary. No, these portions are not actually parabolas.

35.

36.



Summary Exercises on Graphing Circular Functions

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

Section 4.5 Harmonic Motion 47


Section 4.5 Harmonic Motion

1. (a) ( ) 2cos 4 .s t tπ=

(b) ( )1 2;s = the weight is neither moving

upward nor downward. At t = 1, the motion of the weight is changing from up to down.

2. (a) ( ) 45cos .

3s t tπ

=

(b) ( )1 2.5;s =-

upward

3. (a) ( ) 3cos 2.5 .s t tπ=-

(b) ( )1 0;s = upward.

4. (a) ( ) 54cos .

3s t tπ

=-

(b) ( )1 2;s =-

downward

5. ( ) 0.21cos55s t t=

6. ( ) 0.11cos 220 .s t tπ=

7. ( ) 0.14cos110 .s t tπ=

8. ( ) 0.06cos 440 .s t tπ=

9. (a) ( ) 24cos

3s t t

=-

(b) 3.46 units

(c) 1

3 oscillation per second

10. (a) ( ) 6cos2

s t t=-

(b) 2.30 units

(c) 1

4 oscillation per second.

11. (a) ( ) 2sin 2 ;s t t= amplitude: 2; period: ;

frequency 1

= rotation per second

(b) ( ) 2sin 4s t t= amplitude: 2; period: ;

2

frequency 2

= rotation per second

12. (a) 4 in

(b) 1

8t = sec

(c) frequency: 4 cycles per sec; period:1

4 sec

13. period ;4

4

π oscillations per second

14. 2

8 ft

15. 2

1

16. (a) amplitude: 1

;2

period: 2 ; frequency

2

2 oscillatioin per second

(b) ( ) 1sin 2

2s t t=

17. (a) 5 in

(b) 2 cycles per sec; period 1

2 sec

(c) after 1

4 sec.

(d) 4,»

after 1.3 sec, the weight is about 4

in. above the equilibrium position.



18. (a) 4 in

(b) 5

cycles per sec; period

5

sec

(c) after 10

π sec

(d) 2;»

After 1.466 sec, the weight is about 2 in. above the equilibrium position.

19. (a) ( ) 3cos12s t t=-

(b) 6

π sec

20. (a) ( ) 2cos6s t t=-

(b) 3 cycles per sec

21. 0; ; they are the same

22. For 1 2Y and Y /2,2

e ππ -æ ö÷ç ÷ç ÷çè ø; for 1 3Y and Y

none in [0, ] because sin 1,2

π=

2 2sin .2

e eπ ππ- -=

Chapter 4 Review Exercises

1. B

2. D

3. sine; cosine; tangent; cotangent

4. secant; cosecant; tangent; cotangent

5.. 2; 2 ; none; none

6. not applicable; ;3

none; none

7. 1

;2

2

;3

none; none

8. 2; 2

;5

none; none

9. 2; 8 ; up 1 unit; none

10. 1

;4

3 ; up 3 units; none

11. 3; 2 ; none; 2


12. 1; 2 ; none; 3

4

π units to the right

13. not applicable; ; none; 8


14. not applicable; 2; none; 2 units to the right

15. not applicable; ;3

none;

9


16. not applicable; 2 ; none; 3

2

π unit to the left

17. tangent

18. sine

19. cosine

20. cosecant

21. cotangent

22. secant

23. Answers will vary

24. Answers will vary

25.

26.

27.

Chapter 4 Review Exercises 49


28.

29.

30.

31.

32.

33.

34.

35.

36.

37.

38.



39.

40.

41.

42.

43. [−2, 2]

44. ( ] [ ), 2 2,-¥ - ¥

45. y = −sin x + 1

46. 1

cos 22

y x=

47. 1

2 tan2

y x=

48. y = 2 csc x − 1

49. (a) The shorter leg of the right triangle has length 2 1h h- . Thus, we have

( )2 12 1

cot cotd d h h

h hθ θ= = -

-

(b)

50. (a) 12.3 hr.

(b) 1.2 ft.

(c) 1.56 ft

51. (a) 30

(b) 60

(c) 75

(d) 86

(e) 86

(f) 60

52. (a) See the graph in part (d).

(b) ( ) ( )25.5sin 4 47.56


(c) See part b.

(d) Plotting the data with

( ) ( )25.5sin 4 47.56


on the

same coordinate axes gives an excellent fit.

(e)

53. (a) 100

(b) 158

(c) 122

(d) 296

Chapter 4 Test 51


54. (a) about 20 years.

(b) a maximum of about 150,000; a minimum of about 5000.

55. amplitude: 4; period: 2; frequency: 1

2

56. amplitude: 3; period: ; frequency: 1

57. The frequency is the number of cycles in one

unit of time. 4;0; 2 2

58. The period is the time to complete one cycle. The amplitude is the maximum distance (on either side) from the equilibrium point.

Chapter 4 Test

1. (a) secy x= (b) siny x=

(c) cosy x= (d) tany x=

(e) cscy x= (f) coty x=

2. (a) 12

1 cosy x= +

(b) 1

cot2

y x=-

3. (a) ( ),-¥ ¥

(b) [−1, 1]

(c) 2

π

(d) ( ] [ ), 1 1,-¥ - È ¥ .

4. (a)

(b) 6

(c) [−3, 9]

(d) 3-

(e) 4

π unit to the left

5.

6.

7.

8.

9.

10.

11.



12.

13. (a) ( ) ( )16.5sin 4 67.56


(b) 16.5; 12; 4 units to the right; 67.5 units up

(c) 53

(d) 51°F in January; 84°F in July (e) Approximately 67.5° would be an

average yearly temperature. This is the vertical translation.

14. (a) 4 in

(b) after 1

8 sec.

(c) 4 cycles per sec; 1

4 sec

15. The functions y = sin x and y = cos x both have all real numbers as their domains. The

functions ( ) sintan

cos

xf x xx

= = and

( ) 1sec

cosf x x

x= = both have cos x in their

denominators. Therefore, both the tangent and secnt functions have the same restrictions on

their domains. Similarly, ( ) coscot

sin

xf x xx

= =

and ( ) 1csc

sinf x x

x= = both have sin x in

their denominators, and so have the same restrictions on their domains.

lial trig chapter 4 10e - · pdf filesection 4.3 graphs of the tangent and cotangent functions...

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