Chapter 4. Graphs of the Circular Functions. Section 4.1 Graphs of the Sine and Cosine Functions 16.
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1 Chapter 4 Graphs of the Circular Functions Section 4. Graphs of the Sine and Cosine Functions. G. A. E 4. D 5. B 6. H 7. F 8. C 9. D 0. B. C. A Copyright 0 Pearson Education, Inc.
2 Section 4. Graphs of the Sine and Cosine Functions. 7. ;. The graph of y = sin ( x) is the same as the graph of y = sin x because the sine function is an odd function.. 4 ; 8. ; 9. 8 ; 4. ; 0. ; 5. 8 ; 6. 6 ;. ;. ;5 Copyright 0 Pearson Education, Inc.
3 4 Chapter 4 Graphs of the Circular Functions. ; 8. 8; 4. ; 9. ; 5. ; 40. ; 6. ; 7. 4; 4. y= cosx 4. y=- sinx 4. y =- cos x 44. y= cos x 45. y= sin4x 46. y=- cos4x 47. (a) 80 ; 50 (b) 5 (c) about 5,000 yr. (d) downward. 48. (a) 0 mm; 80 mm (b) 0 (c) hours P.M; 0. ft Copyright 0 Pearson Education, Inc.
4 Section 4. Graphs of the Sine and Cosine Functions :9 P.M; 0 ft 5. :8 A.M;.4 feet 54. (a) hr (b) yr. 55. (a) 5; 60 (b) 60 cycles are completed. (c) 5;.545; ; ;.545 (d) 56. (a).8; (a) Cx ( ).04 x.6x 0 7.5sin x (b) Answers will vary. (c) C( x) = 0.04( x-970) + 0.6( x - 970) sin é ( x-970) ù ë û 59. (a) (b) 8 (c) 57 (d) 58 (e) 7 (f) 6 (b) 0 (c) -.074;.74;-.074;-.074; (a) (b).998 watts per m watts per m (d) 57. (a) (c) watts per m (d) Answers may vary. A possible solution is N = 8.5. Other answers are possible. Since N represents a day number, which should be a natural number, we might interpret day 8.5 as noon on the 8 nd day. 4 ;40 or ;0 or 6. No, we can t say that sin bx = bsin x. If b is not zero, then the period of y= sin bx is, b 5 9 (b) Maximums: x =,,, Minimums: x =,,, (c) Answers will vary. and the amplitude is. The period of y= bsin x is, and the amplitude is b. 64. No, we can t say that cosbx = bcos x. If b is not zero, then the period of y= cosbx is, b and the amplitude is. The period of y= bcos x is, and the amplitude is b. Copyright 0 Pearson Education, Inc.
5 6 Chapter 4 Graphs of the Circular Functions 65.X , Y X is cos, and Y is sin.. A. D. The graph of y= sin x+ is the graph of y= sin x translated vertically unit up, while the graph of y= sin( x+ ) is the graph of y= sin x shifted horizontally unit left. 4. y= sin x+ 66. X =, Y ; sin B 6. D 7. C 8. A 9. If the graph of y = cos x is translated units horizontally to the right, it will coincide with the graph of y = sin x. 67. X =, Y ; cos Answers will vary. The x-coordinate is cosθ and the y-coordinate is sin. θ Section 4. Translations of the Graphs of the Sine and Cosine Functions. D. G. H 4. A 5. B 6. E 7. I 8. C 9. C 0. B 0. If the graph of y = sin x is translated units horizontally to the left, it will coincide with the graph of y = cos x.. y = + sin x. y = + cos x. y= cos( x- ) 4. y= cos( x- ) 6 5. ; ; none; units to the left 6. ; ; none; units to the left ;4 ; none; units to the left 4 ;4 ; none; units to the left 9. ; 4, none; 0. ; none;.. unit to the right unit to the right ; ; up units; unit to the right 5 ; ; down unit; units to the right Copyright 0 Pearson Education, Inc.
6 Section 4. Translations of the Graphs of the Sine and Cosine Functions Copyright 0 Pearson Education, Inc.
7 8 Chapter 4 Graphs of the Circular Functions Copyright 0 Pearson Education, Inc.
8 Section 4. Translations of the Graphs of the Sine and Cosine Functions From the sine regression we have y».4sin 0.5x ( ) 58. (a) 7.5 F (b) See the graph in part (d) é ù (c) f ( x) = 9.5cos ( x- 7) êë6 úû (d) The function gives an excellent model for the data. é ù f ( x) = 9.5cos ( x- 7) êë6 úû 57. (a) Yes (b) This line represents the average yearly temperature in Seattle of 5.5 F. (This is also the actual average yearly temperature.) (c).5; ; 4.5 é ù (d) f ( x) =.5sin ( x- 4.5) êë6 úû (e) The functions gives an excellent model for the data (f) (e) From the sine regression we have y= 9.68sin 0.5x ( ) Copyright 0 Pearson Education, Inc.
9 40 Chapter 4 Graphs of the Circular Functions ; ; Chapter 4 Quiz (Section 4. 4.) 7.. 4; ; units up; 4 units to the left. 4;. ; 4. ; 8. y= sin x 9. y= cos x 0. y=- sin x. 7 F. 60ºF; 84ºF Section 4. Graphs of the Tangent and Cotangent Functions. C. A. B 4. D 5. F 6. E 7. Copyright 0 Pearson Education, Inc.
10 Section 4. Graphs of the Tangent and Cotangent Functions Copyright 0 Pearson Education, Inc.
11 4 Chapter 4 Graphs of the Circular Functions Copyright 0 Pearson Education, Inc.
12 Section 4. Graphs of the Tangent and Cotangent Functions domain: ì ï ü íx x¹ ( n+ ), where n is any integer ï ý. ïî ïþ -, range: ( ) sin( -x) -sin x 45. tan( - x) = = =-tan x, cos( -x) cos x ì ï ü íx x¹ ( n+ ), where n is any integer ï ý. ïî 4 ïþ cos( -x) cos x 46. cot( - x) = = =-cot x, sin( -x) -sin x { x x¹ n, where n is any integer }. 47. (a) 0 m (b).9 m. (c). m (d). m (e) t = 0.5 leads to tan, undefined. 48. Answers will vary. which is.. y = tan x 4. y = cot x 5. y = cot x 6. y = tan x 7. y= + tan x 8. y = + cot x 9. True 40. False 4. False 4. True. 4. four x= + n, 4 5. approximately ì ïx x= n, ü í ï ý ïî where n is an integer ïþ Copyright 0 Pearson Education, Inc.
13 44 Chapter 4 Graphs of the Circular Functions Section 4.4 Graphs of the Secant and Cosecant Functions. B. C. D 4. A Copyright 0 Pearson Education, Inc.
14 Section 4.4 Graphs of the Secant and Cosecant Functions y = + csc x. y = + csc x. y = sec x 4. y= - csc x 5. True 6. False 7. True 8. True 9. None ì 0. domain: ï n ü íx x¹, where n is any integerï ý ïî 4 ïþ range: (-, ] È [, ) since a =.. sec( - x) = sec( x) cos( -x) = cos( x) =, ì ï ü íx x¹ ( n+ ), where n is any integer ï ý. ïî ïþ. csc( - x) = csc x sin( -x) = -sin x =-, { x x¹ n, where n is any integer}. (a) 4m (b) 6. m (c) 6.7 m 4. Answers will vary. No, these portions are not actually parabolas For exercises 9 4, other answers are possible. 9. y = sec 4x 0. y = sec x Copyright 0 Pearson Education, Inc.
15 46 Chapter 4 Graphs of the Circular Functions Summary Exercises on Graphing Circular Functions Copyright 0 Pearson Education, Inc.
16 Section 4.5 Harmonic Motion 47 Section 4.5. (a) st () = cos4 t. Harmonic Motion (b) s () = ; the weight is neither moving upward nor downward. At t =, the motion of the weight is changing from up to down. 4. (a) st () = 5cos t. (b) s () =-.5; upward. (a) st () =- cos.5 t. (b) s () = 0; upward. 4. (a) 5 st () =- 4cos t. (b) s () =-; downward 5. st () = 0.cos55 t (c) oscillation per second 0. (a) st () =- 6cos t (b).0 units (c) oscillation per second. 4. (a) st () = sin t; amplitude: ; period: ; frequency = rotation per second (b) st () = sin4t amplitude: ; period: ; frequency = rotation per second. (a) 4 in (b) t = sec 8 (c) frequency: 4 cycles per sec; period: 4 sec 4. period ; oscillations per second 4 6. st () = 0.cos 0 t ft 7. st () = 0.4cos0 t. 8. st () = 0.06cos 440 t. 6. (a) amplitude: ; period: ; frequency oscillatioin per second (b) st () = sin t 7. (a) 5 in (b) cycles per sec; period sec (c) after 4 sec. 9. (a) st () =- 4cos t (d) 4,» after. sec, the weight is about 4 in. above the equilibrium position. (b).46 units Copyright 0 Pearson Education, Inc.
17 48 Chapter 4 Graphs of the Circular Functions 8. (a) 4 in (b) 5 cycles per sec; period sec 5 (c) after 0 sec (d)» ; After.466 sec, the weight is about in. above the equilibrium position. 9. (a) s( t) =- cost (b) sec 6 0. (a) s() t =- cos6 t (b) cycles per sec. 0; ; they are the same æ /. For Y and Y, e - ö ç çè ø ; for Yand Y none in [0, ] because sin =, - - e sin = e. Chapter 4 Review Exercises. B. D. sine; cosine; tangent; cotangent 4. secant; cosecant; tangent; cotangent 5.. ; ; none; none. not applicable; ; none; unit to the right 8 4. not applicable; ; none; units to the right 5. not applicable; ; none; unit to the right 9 6. not applicable; ; none; unit to the left 7. tangent 8. sine 9. cosine 0. cosecant. cotangent. secant. Answers will vary 4. Answers will vary not applicable; ; none; none 7. ; ; none; none ; ; 5 none; none 9. ; 8 ; up unit; none 0. ; 4 ; up units; none. ; ; none; units to the left. ; ; none; units to the right 4 Copyright 0 Pearson Education, Inc.
18 Chapter 4 Review Exercises Copyright 0 Pearson Education, Inc.
19 50 Chapter 4 Graphs of the Circular Functions 9. (b) (a). hr. (b). ft. (c).56 ft 5. (a) 0 (b) 60 (c) 75 (d) 86 (e) 86 (f) [, ] 44. (-,- ] [, ) 5. (a) See the graph in part (d). é ù (b) f ( x) = 5.5sin ( x- 4) êë6 úû (c) See part b. (d) Plotting the data with é ù f ( x) = 5.5sin ( x- 4) on the êë6 úû same coordinate axes gives an excellent fit. 45. y = sin x y= cos x y= tan x (e) 48. y = csc x 49. (a) The shorter leg of the right triangle has length h- h. Thus, we have d cotθ = d = ( h-h) cotθ h -h 5. (a) 00 (b) 58 (c) (d) 96 Copyright 0 Pearson Education, Inc.
20 Chapter 4 Test (a) about 0 years. (b) a maximum of about 50,000; a minimum of about amplitude: 4; period: ; frequency: 56. amplitude: ; period: ; frequency: 57. The frequency is the number of cycles in one unit of time. 4;0; 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. (a) y= sec x (b) y= sin x (c) (e) y= cos x (d) y= tan x y= csc x (f) y= cot x. (a) y= + cos x (b) y=- cot x. (a) (-, ) 9. (b) [, ] (c) (d) (-,-] È [, ). 4. (a) (b) 6 (c) [, 9] (d) (e) unit to the left 4. Copyright 0 Pearson Education, Inc.
21 5 Chapter 4 Graphs of the Circular Functions. é ù. (a) f ( x) = 6.5sin ( x- 4) êë 6 úû (b) 6.5; ; 4 units to the right; 67.5 units up (c) 5 (d) 5 F in January; 84 F in July (e) Approximately 67.5 would be an average yearly temperature. This is the vertical translation. 4. (a) 4 in (b) after 8 sec. (c) 4 cycles per sec; 4 sec 5. The functions y = sin x and y = cos x both have all real numbers as their domains. The sin x functions f ( x) = tan x= and cos x f ( x) = sec x= both have cos x in their cos x denominators. Therefore, both the tangent and secnt functions have the same restrictions on cos x their domains. Similarly, f ( x) = cot x= sin x and f ( x) = csc x= both have sin x in sin x their denominators, and so have the same restrictions on their domains. Copyright 0 Pearson Education, Inc.
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