Chapter 6: Extending Periodic Functions - Weeblyachsprecalc.weebly.com/uploads/1/9/8/2/19823981/cpm_precalculus... · Chapter 6: Extending Periodic Functions Lesson 6.1.1 6-1. a

CPM Educational Program © 2012 Chapter 6: Page 1 Pre-Calculus with Trigonometry

Chapter 6: Extending Periodic Functions Lesson 6.1.1 6-1. a. The graphs of y = sin x and y = 1

2 intersect at many points, so there must be more than one solution to the equation.

b. There are two solutions. From the graph we can see y = !6 and y = 5!

6 . c. It shows where the y-coordinate or sin x = 0.5 . d. x = 4!

3 and x = 5!3 . Students may use unit circle or the graph.

6-2. Draw a vertical line at x = 1

2 . The angles that satisfy the equation are x = !3 and x = 5!

3 . 6-3. A horizontal line drawn at y = 2does not intersect the unit circle. The value 2 is not in the

range for y = sin x . 6-4. Examples of trig equations: cos x = 3 , csc x = 0 Examples of non-trig equations: x2 + 3x + 4 = 0 , 3x + 4 = 2x !1+ x 6-5. a. sin x +1 = 0

sin x = !1

x = 3"2

b. 2 cos x = !1

cos x = ! 12

x = 2"3 ,

4"3

c. cos x = ! 2

x = 3"4 ,

5"4

d. 2 sin x ! 3 = 0

2 sin x = 3

sin x = 32

x = "3 ,

2"3

6-6. a. All real numbers. b. !1 " y " 1 c. The functions both have a period of 2! , so a shift of that size would not affect either

function. 6-7. a. There would be an infinite number of solutions. b. 2 solutions: 0 and π c. Infinitely many. d. An integer multiple of 2! , because it is the period (2!n for n an integer).


6-8. a. There are an infinite number of solutions. There are 12 solutions on the graph given. b. 5!

6 + 2! = 5!6 + 12!6 = 17!

65!6 " 2! = 5!

6 " 12!6 = " 7!6

c. Add 2!n to !6 , 5!

6 , n is any integer.

d. !6 + 4! = !

6 +24!6 = 25!

65!6 + 4! = 5!

6 + 24!6 = 29!

6

e. ! "6 ! 5" = ! "

6 !30"6 = ! 31"

6

! 5"6 ! 5" = ! 5"

6 ! 30"6 = ! 35"

6

6-9. a. The y-coordinates of the points are 12 . b. Answers vary, but going around the circle 2! would take us back to the same place as

!6 or! 5!

6 . 6-10. a. 2 sin x ! 3 =

2 sin x = 3

sin x = 32

x = "3 ,

2"3

b. !3 + 2!n,

2!3 + 2!n

Review and Preview 6.1.1 6-11. a. Since the string is 30 inches in length, the maximum point will be 30 inches above the

minimum. b. 30

2 = 15 c. 15 + 5 = 20

d. 2!2.5 =

2!5 2 = 2! " 25 =

4!5 e. ! cos x

f. h = !15 cos 4"5 (t)( ) + 20

6-12. a. csc 5!6 = 1

sin5! 6 =11 2 = 1 "

21 = 2

b. tan !2 =

sin! 2cos ! /2 =

10 " undefined

c. cot 5!3 = cos5! 3sin5! 3 =

1 2" 3 2

= 12 #

2" 3

= 1" 3

= " 33

d. sec 7!6 = 1sin 7! 6 =

1" 3 2

= 1 # 2" 3

= 2" 3

= "2 33


6-13.

a. x0 = 1x1 = 2x2 = 3

b. ! 12

k=1

3

" (k !1)2 + 7

left-sum = ! 12 (1!1)

2 + 7"# $% + ! 12 (2 !1)

2 + 7"# $% + ! 12 (3!1)

2 + 7"# $% = 7 +132 + 5

6-14.

x = b ! y3(b ! y) ! 2y = a3b ! 3y ! 2y = a

!5y = a ! 3b

y = 3b ! a5

x = b ! 3b ! a5

x = 5b5

! 3b ! a5

= a + 2b5

6-15. 4

5 =x32

128 = 5x1285 = x

x = 25.6 feet

6-16. a. 9!

0.2 =36!x

9! " x = 36! "0.29! x = 7.2!

x = 7.29 = 0.8 liters/hr

b.

6-17. g(!1) = (!1)2 ! 2(!1)

g(!1) = 1+ 2 = 3

g(3) = 32 ! 2(3)g(3) = 9 ! 6 = 3

g(a) = a2 ! 2(a)g(a) = a2 ! 2a

g(t ! 2) = (t ! 2)2 ! 2(t ! 2)g(t ! 2) = t2 ! 4t + 4 ! 2t + 4

= t2 ! 6t + 8

V = 13 ! "62 "10 = 120!

12 "V = 1

2 "120! = 60!

60! = 13 ! r

2 "h

180 = r2 53 r( )

108 = r3

r = 4.76229!0.2 =

4.76222!x

9! " x = 22.679! "0.29! x = 4.536!

x = 4.5369 = 0.504 liters/hr


6-18. a. x2

x2 !x! 1x2 !x

= 3

x2 !1x(x!1) = 3

(x+1)(x!1)x(x!1) = 3

x+1x = 33x = x +12x = 1

x = 12

b. xx2 !25

+ x!5x2 !25

= 1

2x!5x2 !25

= 1

2x ! 5 = x2 ! 250 = x2 ! 2x ! 200 = x2 ! 2x +1!1! 2021 = (x !1)2

± 21 = x !1

x = 1± 21

Lesson 6.1.2 6-19. b. Inverses are symmetric about the line y = x . c. No, because it does not pass the vertical line test. 6-20. a. ! "

2 ,"2#$ %&

b. The domain of y = sin!1 x will be the range of y = sin x , so the domain is !1,1[ ] . 6-21. a. !

3 = 1.047 b. It is not in the range of y = sin!1 x . The inverse of sine only selects one of the infinitely

many solutions to the equation. c. x =

!3 + 2!n or 2!3 + 2!n

d. You have to use the unit circle or a wave. 6-22. a. It does not pass the vertical line test. b. 0,![ ] c. The domain of y = cos!1 x is the range of y = cos x , which is !1,1[ ] . The range of

y = cos!1 x is 0,![ ] .


6-23. y = sin!1(x) :!!D : !1,1[ ] ,!R : ! "

2 ,"2#$ %& y = cos!1(x) :!!D : !1,1[ ] ,!R : 0,"[ ]

6-24. a. 0.305 b. ! " 0.305 = 2.837 c. 0.305 + 2!n, 2.837 + 2!n , for n an integer. 6-25. a. vertical line b. 1.266 c. 5.017 = 2! "1.266 d. 1.266 + 2!n, 5.0177 + 2!n or ±1.266+2!n , n an integer Review and Preview 6.1.2 6-26. a. It is not in the range of y = cos!1 x . cos!1 x selects only one of the infinitely many

solutions to the equation. b. x =

!3 + 2!n or 5!3 + 2!n

c. You have to draw and think. 6-27.

tan x = sin xcos x = 0 ! sin x = 0

x = " , 2" , 3" , 4"…x = n" , n is any integer

6-28. a. The equation cos x = !0.3 will have multiple solutions. b. Sylvie needs to include all the solutions, which she can get using a graph or unit circle.

She needs to add multiples of 2π, and include the negative values. x = ±1.875 + 2!n , where n is an integer.

x

y

1 –1

!2

! "2

x

y

1 –1

!

!2


6-29. See diagram at right. a. ! "

3

b. !4

c. 3!4

d. !6

6-30. 22 + x2 = 32

x2 = 5

x = ± 5

cos! = " 5

3 6-31. a.

62 = 102 + 82 ! 2(8)(10) cos x36 = 164 !160 cos x

!128 = !160 cos x0.8 = cos x

cos!1 0.8 = cos!1(cos x)

x = 36.9!

b.

xsin 60!

= 28sin 70!

x sin 70! = 28 sin 60!

0.9397x = 24.2587x = 25.8

6-32. a. log2 1

64( ) = log2 64!1( )= log2 26( )!1 = log2 2!6 = !6

b. log8 1 = 0

c. log8 81 = 1 d. log2(64) = log2(26 ) = 6

e. impossible f. log5 251 3( ) = log5 52( )1 3( )= log5 52 3( ) = 2

3

6-33.

a. 2x3y2 !4x2y2 +2xy2

3xy3!3y3= 2xy2 (x2 !2x+1)

3y3(x!1)

= 2x(x!1)(x!1)3y(x!1)

= 2x(x!1)3y

b. (x+h)2 !x2h = x2 +2xh+h2 !x2

h

= 2xh+h2h

= h(2x+h)h = 2x + h


6-34. a. ! f (x) + 2 b. 2 f (!x)

Flipped over x-axis and up 2. Flipped over y-axis and stretched vertically.

c. 1

f (x) Asymptotes at x = !2, 0 , and 2. Lesson 6.3.1 6-35. The Law of Sines calculation results in the sine of the angle at Icy’s being greater than 1.

The Law of Cosines calculation yields a quadratic equation with no real solutions. 6-36. a.

20sin 28!

= 30sin I

30 !0.4695 = 20 sin I14.08520 = sin I

sin"1 0.70425 = sin"1 sin I44.8! = #I

b.

!D = 180! " 28! " 44.8! = 107.2!20

sin 28!= dsin 107.2

d #0.4695 = 20 sin107.2!

d = 19.20560.4695

d = 40.69 m

(or !I = 135.2! , but don ot point this out yet) c. Katya missed the possibility that !I could be obtuse.

!I = 180! " 44.8! = 135.2!

!D = 180! "135.2! " 28! = 16.8!20

sin 28!= dsin 16.8!

d #0.4695 = 20 sin16.8!

d = 5.78060.4695

d = 12.31 m

x

y

x

y

x

y


6-37. a. See diagram at right. The horizontal line crosses the unit

circle at two different angles. b. Inverse sine has a restricted range, which does not

include the 2nd quadrant. 6-38. a.

10sin 90!

= asin 30!

12 !10 = a !1

a = 5

b.

10sin C = 5

sin 30!

10 ! sin 30! = 5 ! sinC5 = 5 sinC1 = sinC"C = 90!

c.

10sin C = 3

sin 30!

10 ! sin 30! = 3 ! sinC5 = 3 sinC53 " sinCNot possible since the range of sineis #1,1[ ] .

d.

10sin C = 7

sin 30!

10 ! sin 30! = 7 ! sinC5 = 7 sinC57 = sinC

"C = sin#1 57( )

"C = 45.58!

or "C = 180! # 45.6! = 134.4!

e. !ACB = 180! " !BC #C = 180! " !B #C C since !BC "C is isosceles. f. Supplementary angles have the same sine. g. One triangle. 6-39. 0 triangles if a < c sin A ; 1 triangle if a = c sin A or a ! c , 2 triangles if c sin A < a < c . Review and Preview 6.3.1 6-40.

9sin 34!

= 8sin C

8 ! sin 34! = 9 ! sinC4.47 = 9 sinC4.479 = sinC

"C = sin#1 4.479( )

"C = 29.8!

!B = 180! " 34! " 29.8! = 116.2!9

sin 34!= ACsin 116.2!

AC sin 34! = 9 # sin116.2!

AC = 8.07530.5592 = 14.44 cm

There is only one solution to the triangle since ∠C must be smaller than ∠B (since 8 < 9). Therefore, ∠C cannot be obtuse and there can only be one solution.


6-41. a. sin x = 4

5

sin!1 45( ) = 0.927

b. x = 0.927 and ! " 0.927 = 2.214

c. 0.9273+ 2pn, 2.2143+ 2pn , n is an integer. 6-42. g(x) = k

x2

1.2 = k42

k = 16 !1.2 = 19.2

g(6) = 19.262

g(6) = 965 ! 136 =

85 !

13 =

815

g(!3) = 19.2(!3)2

g(!3) = 965 " 19 =

325 " 13 =

3215

6-43. y = 1+ x

x+2

x = 1+ yy+2

x = y+2+yy+2

x(y + 2) = 2y + 2xy + 2x = 2y + 2xy ! 2y = 2 ! 2xy(x ! 2) = 2 ! 2x

y = 2!2xx!2 = 2x!2

2!x

f !1(x) = 2x!22!x

6-44. 1

g(x) =1

x(x+2)(x!3)

Asymptotes occur when the denominator equals zero. This occurs when x = 0, !2, 3 . 6-45. 1+cos!

(1"cos! )(1+cos! ) +1"cos!

(1"cos! )(1+cos! ) =1+cos!+1"cos!

1"cos2 != 2sin2 !

= 2 csc2 !

6-46.

f (x) = 27(9)12 x!1 = 333

2 12 x!1( ) = 333x!2 = 33+x!2 = 3x+1 = 3(3)x

6-47.

f (x ! 3) ! 2 =!2(x ! 3) + 3! 2 for x < 1! 3

2(x ! 3) !1 ! 2 for x " 1! 3#$%

&%

h(x) =!2(x ! 3) +1 !!for x < !2

2(x ! 3) !1 ! 2 for x " !2#$%

&%


Lesson 6.1.4 6-48. a. You would find vertical asymptotes when cos x = 0 . These occur at x = ! 3"

2 , !"2 ,

"2 ,

3"2 .

b. This would be when the graph of tan x crosses the x-axis, which are the roots, and they occur at x = 2! , "! , 0,! , 2! .

6-49. a. x ! n"

2 , where n is any odd integer. b. All real numbers.

c. y = 0, x = +!n , n is any integer. d. x = n!2 , where n is any odd integer.

6-50. a. Restrict the range. b. Range: ! "

2 ,"2#$ %&

6-51. a. lim

x!"tan#1(x) = $

2 b.

limx!"#

tan"1(x) = " $2

6-52. tan! = opposite

adjacent =yx

6-53.

tan! = 12

tan"1 tan! = tan"1 12( )

! = 26.6! or 0.464 radians

6-54.

adjacent side = 452 = 22.5

tan! = oppositeadjacent =

822.5

tan"1 tan! = tan"1 822.5( )

! = 19.573!

6-55. ! = 1.2 radians

tan1.2 = 2.572approximate slope = 2.572


Review and Preview 6.1.4 6-56. a. 2 sin x !1 = 0

2 sin x = 1

sin x = 12

x = "6 + 2"n, 5"

6 + 2"n, n is an integer

b. 2 + 2 cos x = 02 cos x = !2

cos x = !22 = !1

x = " + 2"n, n is an integer

c. 2 ! 2 sin x = 0

!2 cos x = ! 2

cos x = 22

x = "4 + 2"n, 3"

4 + 2"n, n is an integer

d. cos x + 3.8 = 0cos x = !3.8cos x />1 " no solution

6-57. Yes, the first is the inverse function, the second the reciprocal function of y = cos x . 6-58. sin x = 0.3 has infinite solutions unless we are working with a restricted values of x. The

expression sin!1 0.3 = x has only one solution when sin!1 x is a function. 6-59. It is false. For example, take a = !

6 , b = !3 . sin !

3 +!6( ) = sin 2!

6 + !6( ) = sin !

2( ) = 1but sin !

3( ) + sin !6( ) = 3

2 + 12 =

3+12 " 1

6-60. 2x2 !+ 8x + a = 2(x2 +2xb + b2 )!

2x2 !+ 8x + a = 2x2 +4xb + 2b2

8 = 4bb = 2a =!2b2 = 2 !22 = 8

6-61. Amp. = 3, horizontal shift = 2 to the right, vertical shift = 1 up, period = 2!! 2 =

2!1 " 2! = 4 .

6-62.

tan 2!3 = sin2! 3cos2! 3 =

3 2"1 2 = 3

2 # " 21 = " 3


6-63. a. slope of PR = 2!6

14!(!4) =!418 = ! 2

9

perpendicular slope = 92

midpoint of PR = !4+102 , 6+2

2( ) = (5, 4)

y ! 4 = 92 (x ! 5)

y = 92 (x ! 5) + 4

b. slope of median = 12!42!5 = 8

!3 = ! 83

y ! 4 = ! 83 (x ! 5)

y = ! 83 (x ! 5) + 4

c. slope of PR = 2!614!(!4) =

!418 = ! 2

9

perpendicular slope = 92

y !12 = 92 (x ! 2)

y = 92 (x ! 2) +12

6-64. x0 = 1.25, x1 = 1.75, x2 = 2.25, x3 = 2.75,!x4 = 3.25, x5 = 3.75, x6 = 4.25, x7 = 4.75

xk = 0.5k +1.25

sum = 12

10.5k+1.25

k=0

7

! " 1.600

Lesson 6.2.1 6-65. Laurel is. Hardy’s equation only shifts the graph !6 to the right since

H (x) = sin 3x ! "2( ) = sin 3 x ! "

6( )( ) . 6-66. a. x = !

2 b. x = !6 c. H (x) = sin 3x ! "

2( ) = sin 3 x ! "6( )( )

6-67. y = 2 sin(3(x ! " )) + 4 6-68.

a. Amplitude = !1! (!5)2

= 2

Horizontal shift is !2 to the right. Vertical shift is 3 down. The period is 2!2 = ! .

b. y = 2 sin 2 x ! "2( )( ) ! 3


6-69. a. y = 3 cos(! (x +1)) " 2 b. y = 2 sin 1

3 x ! "2( )( )

6-71. a. (0.4, 46) and (2.2, 26)

b. Period = 2(2.2 ! 0.4) = 3.6 , Amplitude = 26 ! 462

= 202

= 10 , horizontal shift 0.4 or –1.4,

Vertical shift = 26 + 202 = 26 +10 = 36 .

c. One possible answer is h(t) = 10 cos 2!3.6( ) (t " 0.4)( ) + 36.

Review and Preview 6.2.1 6-72. y = 3sin !

2 (x " 2)( ) +1 6-73. 52 + (leg b)2 = 82

(leg b)2 = 64 ! 25

leg b = 39

a. sin! = 58

b. cos! = " 398

c. tan! = 5 8" 39 8

= 58 # "

839

= " 539

# 3939

= " 5 3939

6-74. a. The range of sine and cosine is !1 " y " 1 . b. A fraction can equal 37 without the numerator being 3 and the denominator being 7. For

example, 0.30.7 =37 .

c. tan!1 tan x = tan!1 37( )

x = 0.405 or! 0.405 + " = 3.546

x

y

2

–2

–4

4

2 4


6-75. a. x2 ! 4x ! 21 = 0

(x + 3)(x ! 7) = 0 x = !3, 7

b. (x ! 2)(x +1) = 4x2 ! x ! 2 = 4x2 ! x ! 6 = 0

(x ! 3)(x + 2) = 0 x = !2, 3

c. 3x2 + x = 103x2 + x !10 = 0

(3x ! 5)(x + 2) = 0

x = 53 , !2

d. 6x2 + 5x = 256x2 + 5x ! 25 = 0

(3x ! 5)(2x + 5) = 0

x = 53 , ! 5

2

6-76.

tan x!csc xsec x =

sin xcos x !

1sin x1

cos x=

1cos x1

cos x= 1

6-77.

tan 28! = 0.532y ! 912 = ±0.532(x ! 285)

6-78.

sec x!tan xsin x =

1cos x !

sin xcos x

sin x =sin xcos2 xsin x = sin x

cos2 x! 1sin x =

1cos2 x

= sec2 x

6-79. a. y = !3 cos(2x) !1 b. y = 2 sin x + !

4( ) " 2

c. y = sec(x) d. y = tan!1 x 6-80. h = kV

r2

15 = k204

60 = 20kk = 3

h = 3Vr2

h = 3!109

h = 309 = 10

3


Lesson 6.2.2 6-81. a. cos x ! "

4( ) = cos x sin "4( ) + sin x cos "

4( )

b. cos x ! "4( ) = cos x # 2

2 + sin x # 22

c. cos x ! "4( ) = 2

2(cos x + sin x)

2 cos x ! "4( ) = cos x + sin x

d. 2 6-82. a.

cos(90! -!) = cos 90! cos! + sin 90! sin!= 0 " cos! +1 " sin!= sin!

b.

sin(90! -!) = sin 90! cos! " cos 90! sin!= 1 # cos! + 0 # sin!= cos!

c. cot! = cos!sin! = sin(90º"! )

cos(90º"! ) = tan(90º "!)

d. csc! = 1

sin! = 1cos(90!"! )

= sec(90! "!)

6-83. a. cos! = " 3

5

b. sin ! = " 74

c. sin(! " #) = 45 $ " 3

4( ) " " 35( ) $ " 7

4( ) = " 1220 "

3 720 = "12"3 7

20

d. cos(! + ") = # 35 $ # 3

4( ) # 45( ) $ # 7

4 = # 920 + 4 7

20 = 9+4 720

Review and Preview 6.2.2 6-84. a. 20 b. x-coordinate: 15 !11.31 = B !15!!!"!!B = 30 !11.31 = 18.69 (18.69, 5) c. x-coordinate: 11.31! 5 = 5 ! C !!!"!!C = 10 !11.31 = !1.31 (–1.31, 5)


6-85. a. Amplitude is 10.

Horizontal shift is 5 to the right. Vertical shift is 24 up. The period is 2!! 2 = 2! " 2! = 4 .

b. See graph at right. 6-86. a. 10 sin !

2 x " 5( )( ) + 24 = 20

If u = !2 (x " 5)

10 sin u = "4

sin u = " 25

sin"1 sin u = sin"1 " 25( )

u = !0.4115"2 (x ! 5) = !0.4115x ! 5 = !.262x = 4.738 ! 4 = 0.738

b. y = 10 sin !2 x " 5( )( ) + 24

x = 3.262

6-87. cos(! -") = cos! cos" + sin ! sin" = #1cos" + 0 $ sin" = # cos" 6-88. sin(! -") = sin ! cos" # cos! sin" = 0 $ cos" + #(#1) $ sin" = sin" 6-89. a. !

2 ,3!2 b. 2 sin x + 2 = 0

2 sin x = ! 2

sin x = ! 22

x = 5"4 ,

7"4

c. cos x 2 sin x + 2( ) = 0

cos x = 0 or 2 sin x + 2 = 0

x = !2 , 3!

2 , 5!4 , 7!

4

d. cos x 2 sin x + 2( ) = 0

cos x = 0 or 2 sin x + 2 = 0

x = !2 + !n, 5!

4 + 2!n, 7!4 + 2!n

6-90.

(csc x + cot x)(1! cos x) = 1

sin x +cos xsin x( ) (1! cos x) =

1+cos xsin x( ) (1! cos x) = 1!cos2 x

sin x = sin2 xsin x = sin x

6-91.

x2y!3 + x!2yy!1 + x!2

= x2y3 " x2y!3 + x2y3 " x!2yx2y3 " y!1 + x2y3 " x!2

= x4 + y4

x2y2 + y3


Lesson 6.2.3 6-92. y = 20 cos !6 (x " 2) + 44

Amplitude: 64 ! 242

= 402

= 20 inches Period: 12 = 2!b !!"!!b =

!6

Horizontal shift: 2 (hours) to the right Vertical shift: 44 (inches) up

a. y = 20 cos !6 ("0.5 " 2) + 44y = 20 cos("1.309) + 44y = 5.1764 + 44y = 49.18 inches

b. 2’7” tall = 31 inches

31 = 20 cos !6 (x " 2) + 44

" 1320 = cos !

6 (x " 2)

cos"1 " 1320( ) = !

6 (x " 2)

4.3514 = x " 2x = 6.3514 # 6 hours 21 minutes2 pm " 6 hours 21 minutes # 9 : 39 a.m.

6-93. 1. h = 34 cos! (t "1.25) + 34

Amplitude: 68 ! 02

= 34 inches Period: 2 = 2!b !!"!!b = !

Horizontal shift: 1.25 (seconds) to the right Vertical shift: 34 (centimeters) up a. h = 34 cos! (15.6 "1.25) + 34

h = 34 cos(45.082) + 34h = 15.4357 + 34h = 49.44 cm

b. 12 = 34 cos! (x "1.25) + 34

" 2234 = cos! (x "1.25)

cos"1 " 2234( ) = ! (x "1.25)

2.2745! = x "1.25

0.724 = x "1.25x = 1.974 secx = 1.25 " 0.724 = 0.526 sec

2. h = 4 cos 2!3 (x - 2)( ) + 5

Amplitude: 9 !12

= 4 feet Period: 3 = 2!b !!"!!b =

2!3

Horizontal shift: 2 (seconds) to the right Vertical shift: 5 (feet) up

a. h = 4 cos 2!3 (5.4 " 2) + 5h = 4 cos(7.1209) + 5h = 4 #0.6691+ 5h = 6.677 ft

b. 7.2 = 4 cos 2!3 (x " 2) + 5

0.55 = cos 2!3 (x " 2)

cos"1(0.55) = 2!3 (x " 2)

0.4719 = x " 2x = 2.472 secx = 2 " 0.4719 = 1.528 sec


3. d = 29 sin !3 (t " 5.5)( ) + 54

Amplitude: 83! 252

= 29 cm Period: 6 = 2!b " b = !

3

Horizontal shift: 5.5 (seconds) to the right Vertical shift: 54 (centimeters) up

a. h = 29 sin !3 (8 " 5.5) + 54

h = 29 sin(2.618) + 54h = 14.5 + 54h = 68.5 cm

b.

4. A = 1.1 sin !

3 t – 3.5( )( ) +1.7

Amplitude: 2.8 ! 0.62

= 1.1 liters Period: 6 = 2!b !!"!!b =

!3

Horizontal shift: 3.5 (seconds) to the right Vertical shift: 1.7 (liters) up a. A = 1.1sin !

3 (3.5 " 3.5)( ) +1.7

A = 1.1sin(0) +1.7A = 1.7 liters

b. 2.3 = 1.1sin !3 (t " 3.5) +1.7

0.5455 = sin !3 (t " 3.5)

sin"1(0.5455) = !3 (t " 3.5)

0.5509 = t " 3.5t = 4.051 seconds

5. h = 23 cos 8!3 (x " 0.125)( ) + 38

Amplitude: 76 ! 302

= 23 cm Period: 34 =2!b !!"!!b = 2! # 43 =

8!3

Horizontal shift: 0.125 (seconds) to the right Vertical shift: 38 (cm) up a. h = 23 cos 8!

3 (5.2 " 0.125)( ) + 38h = 23 cos(42.5162) + 38h = 23 #0.1045 + 38h = 40.404 cm

b. 59 = 23 cos 8!3 (x " 0.125)( ) + 38

59 = 23 cos 8!3 (x " 0.125)( ) + 38

cos"1 2123( ) # 38! = x " 0.125

x = 0.075

6. F = 19 sin !12 (t -10)( ) + 84

Amplitude: 103! 652

= 19 degrees Period: 24 = 2!b !!"!!b =

!12

Horizontal shift: 10 (hours) to the right Vertical shift: 84 (degrees) up a.

F = 19 sin !12 (11"10)( ) + 84

F = 19 sin !12( ) + 84

F = 4.918 + 84F = 88.918!

b. 98 = 19 sin !12 (t "10)( ) + 84

14 = 19 sin !12 (t "10)( )

sin"1 1419( ) = !

12 (t "10)

3.164 = t "10, t = 13.164

1.164 hours after noon or about 1:10 p.m.

33 = 29 sin !3 (t " 5.5) + 54

"2129 = sin !

3 (t " 5.5)

sin"1 "2129( ) = !

3 (t " 5.5)

"0.7733 = t " 5.5t = 4.7267 seconds


7. h = 15.5 sin 5!2 (t " 3.4)( ) + 23.5

Amplitude: 39 ! 82

= 15.5 cm Period: 45 =2!b !!"!!b =

10!4 = 5!

2

Horizontal shift: 3.4 (seconds) to the right Vertical shift: 23.5 (centimeters) up

a. h = 15.5 sin 5!2 (15 " 3.4)( ) + 23.5

h = 15.5 sin(0) + 23.5h = 23.5 cm

b. 13 = 15.5 sin 5!2 (t " 3.4)( ) + 23.5

" 10.515.5 = sin5!2 (t " 3.4)( )

sin"1 " 10.515.5( ) = 5!2 (t " 3.4)

"0.0948 = t " 3.4t = 3.3052

Subtracting four periods from this (0.8 ! 4 = 3.2 ) gives 3.3052 ! 3.2 = 0.105 seconds. 8. h = 31sin(p(t ! 3.5)) + 71

Amplitude: 62 ! 02

= 31 cm Period: 2 = 2!b !!"!!b = !

Horizontal shift: 3.5 (seconds) to the right Vertical shift: 71 (cm) up a. h = 31sin(! (20 " 3.5)) + 71

h = 31sin(51.8363) + 71h = 31+ 71 = 102 cm

b. 52 = 31sin(! (t " 3.5)) + 71

" 1931 = sin(! (t " 3.5))

sin"1 " 1931( ) = ! (t " 3.5)

"0.21 = t " 3.5t = 3.29 seconds

9. h = 6 cos !4 (t " 5)( ) +12

Amplitude: 18 ! 62

= 6 cm Period: 8 = 2!b !!"!!b =

!4

Horizontal shift: 5 (seconds) to the right Vertical shift: 12 (cm) up

a. h = 6 cos !4 (26 " 5)( ) +12

h = 6 cos(16.4934) +12h = "4.2426 +12 = 7.757 cm

b. 16 = 6 cos !4 (t " 5)( ) +12

46 = cos !

4 (t " 5)( )cos"1 2

3( ) = !4 (t " 5)

1.0709 = t " 5, t = 6.0709 seconds


Review and Preview 6.2.3 6-94. a. Amplitude is 4. b. Horizontal shift is !2 . Vertical shift is 2. y = 2 + 4 cos x ! "

2( ) y = 2 + 4 sin x Other answers are possible. 6-95. a. 5

3

b. ! 154

c. 23 !

14 +

53 ! " 15

4 = 212 + " 5 3

12 = 2"5 312

d. 53 ! 14 +

23 ! "

154 = 5

12 + " 2 1512 = 5"2 15

12 e. –0.459 6-96.

sin! = "35 = " 3

5

tan! = "3"4 =

34

csc! = 5"3 = " 5

3

sec! = 5"4 = " 5

4

cot! = "4"3 =

43

6-97. a. 2!

3 b. 2!! = 2 c. 2!

! 5 = 2! " 5! = 10 d. 2!! 5 = 2! " 5! = 10

6-98. cos !

2 +"( ) = cos !2 cos" # sin !2 sin"

= 0 $ cos% + #1 $ sin"= # sin"

6-99. 4 cos2 x = 3

cos2 x = 34

cos x = ± 32

x = !6 + !n,

5!6 + !n


6-100.

x !1 x2 ! 3x + Ax2 ! x

! 2x + A!2x ! 2

A ! 2

x ! 2 A ! 2 = 0

A = 2 x ! 2 = x + B

B = !2

6-101. a. See graph at right. b. f (x) = x, g(x) =

1x , h(x) = x +

1x =

x2 +1x

6-102. 4 + 3 cos2 z

4 + 3(1! sin2 z)4 + 3! 3sin2 z7 ! 3sin2 z

6-103. a. log3 x+9

x( ) + log5 52 = 4log3 x+9

x( ) + 2 = 4log3 x+9

x( ) = 2x+9x = 32

9x = x + 98x = 9

x = 98

b. 500(1.15)2x!1 +1000 = 10000

500(1.15)2x!1 = 9000

(1.15)2x!1 = 18

2x !1 = log1.15 18 =log 18log 1.15

2x = 21.6807x = 10.8403

x

y


Lesson 6.3.1 6-104. a. sin(! +! ) b. sin(2! ) = sin(! +! ) = sin! cos! + sin! cos! = 2 sin! cos! 6-105. a. cos(! +! ) b. cos(2! ) = cos(! +! ) =

cos! cos! " sin! sin! =

cos2 ! " sin2 !

c. cos 2! = cos2 ! " sin2 ! =

cos2 ! " (1" cos2 ! ) =cos2 ! "1+ cos2 ! = 2 cos2 ! "1

d. cos 2! = cos2 ! " sin2 ! =

(1" sin2 ! ) " sin2 ! =

1" 2 sin2 !

6-106. a. 2 sin 3x cos 3x = sin(2 ! 3x) = sin 6x b. cos2 40! + sin2 40! = 1 c. cos

2 40! ! sin2 40! = cos(2 " 40!) = cos(80!) d. 1! 2 sin2(y ! 5) = cos(2(y ! 5)) = cos(2y !10) e. sin 30

! cos 40! + cos 30! sin 40! = sin(30! + 40!) = sin(70!) f. 2 cos2(2w) !1 = cos(2 "2w) = cos(4w) 6-107. a. sin x cos x = 1

4

2 ! sin x cos x = 14 !2

2 sin x cos x = 12

b. sin(2x) = 12

sin!1(" ) = sin!1 12( )

" = #6 + 2#n,

5#6 + 2#n

c. 2x = !6 + 2!n,

5!6 + 2!n

x = !12 + !n,

5!12 + !n


6-108. a. 2 cos2 ! = cos 2! +1

cos2 ! = cos 2!+12

b. ! = 2"

" = !2

c. cos2 !2( ) = cos 2 "

!2( ) +12

= cos(!) +12

cos(!2 ) = ± cos(!) +12

d. cos 2! = 1" 2 sin2 !2 sin 2! = 1" cos 2!

sin2 ! = 1" cos 2!2

sin! = ± 1" cos 2!2

sin(#2 ) = ± 1" cos#2

Review and Preview 6.3.1 6-109. sin 2x = 2 sin x cos x = 2 ! " 3

5 ! "45 =

2425

cos 2x = 2 sin2 x "1 = 2 ! " 45( )2 "1 = 32

25 "1 =725

sin x2 =

1"cos x2 =

1"(" 45 )2 =

952 = 9

10 = 310

cos x2 = " 1+cos x2 = "

1+(" 45 )2 = "

152 = " 1

10

6-110. 3sin x = 1

sin x = 13

sin!1 sin x = sin!1 13( )

x1 = 0.340!!!!!x2 = " ! 0.340 = 2.802

6-111. a. sin(2 !5p) = sin10p b. ! sin "

4 !"6( ) = ! sin 3"

12 !2"12( ) = sin ! "

12( ) 6-112. See graph at right. (x !1)(x + 3) " 0 for! ! 3 " x " 1

x

y


6-113. a. Any length such that 4.226 < AT < 10 .

The smallest !A = 0!"!AT = 10 . The largest !A = 155!"!!T = 0!" AT = 4.226 . b. AT = 4.226!or !AT ! 10 c. AT < 4.226 6-114. sin! = 4

5 !!!!!cos " = 513 !!!!!sin " = # 1213

sec(! + ") = 1cos(!+" ) =

1cos! cos "#sin! sin "

= 135 $(5 13)#(4 5)(#12 13)

= 115 65+48 65 =

163 65 =

6563

6-115.

a. cos(1.2) + cos(0.3+1.2) + cos(0.6 +1.2) +…+ cos(2.7 +1.2) = cos(0.3k +1.2)

k=0

9

!

b. 1 – 3+ 5 – 7 + 9 – 11 +!+ 201 = 1+ (!1) " 3+ (!1)2 "5 + (!1)3 " 7 +!+ 201 = (!1)n

n=0

100

# (2n +1)

6-116. a. 2x2 ! x ! 3 = 0

(2x ! 3)(x +1) = 02x ! 3 = 0!!or !!x +1 = 0

x = 32 !!or !!x = !1

b. 2(1! y2 ) + y +1 = 02 ! 2y2 + y +1 = 02y2 ! y ! 3 = 0!!"!!Same answer as part (a).

Lesson 6.3.2 6-117. a. 2 cos2 x + sin x +1 = 0

2(1! sin2 x) + sin x +1 = 0

2 ! 2 sin2 x + sin x +1 = 0

!2 sin2 x + sin x + 3 = 0

2 sin2 x ! sin x ! 3 = 0

b. 2 sin2 x ! sin x ! 3 = 0u = sin x

2u2 ! u ! 3 = 0(2u ! 3)(u +1) = 0

2u ! 3 = 0 or u +1 = 0

u = sin x = 32 or u = sin x = !1

c. sin x = 32 is impossible since 1.5 is greater than 1. sin x = !1!!"!!x = 3#

2


6-118. a. 8c2 ! 4c = 0

4c(2c !1) = 04c = 0 or 2c !1 = 0

c = 0 or c = 12

b. s2 + s ! 2 = 0(s + 2)(s !1) = 0

s + 2 = 0 or s !1 = 0s = !2 or s = 1

6-119. a. 8 cos2 x = 4 cos x

8 cos2 x ! 4 cos x = 04 cos x(2 cos x !1) = 0

cos x = 0 or 2 cos x !1 = 0

x = "2 , 3"

2 or cos x = 12

x = "3 , 5"

3

b. sin2 x + sin x ! 2 = 0(sin x + 2)(sin x !1) = 0

sin x !1 = 0 or sin x + 2 = 0 sin x = 1 or sin x = !2

x = "2 !!!!!!!!!!!!!!!!!!

6-120. sin(x + ! ) + cos(x + ! ) = " cos x

sin x cos! + cos x sin ! + cos x cos! " sin x sin ! = " cos x" cos x + sin x = " cos x

sin x = 0x = n!

6-121. a. The range of cos x is !1 " x " 1. b. You cannot divide cos 2x by cos x and you cannot cancel cos x in the expression 2+cos x

cos x . c. 2 cos2 x ! cos x ! 3 = 0

(2 cos x ! 3)(cos x +1) = 02 cos x ! 3 = 0 or cos x +1 = 0

cos x = 32 or cos x = !1

Solutions: x = " + 2"n

6-122. a. sin x = 0 or cos x = !1

x = 0," , 2" b. x = ! "

3

c. cos x = 0 or tan x = !1

x = "4 , "

2 , 5"4 , 3"

2 , all + 2"n

d. tan x = ! 3

x = 2"3 , 5"

3


Review and Preview 6.3.2 6-123. y = 2x+5

x!2 = 2(x!2)+9x!2 = 2 + 9

x!2 Asymptotes at x = 2 and y = 2 . 6-124. a. cot x

sin x (sec x ! cos x) = 1cos xsin x "

1sin x( ) 1

cos x !cos x1( ) =

cos xsin2 x( ) 1!cos2 x

cos x( ) =1!cos2 xsin2 x

= sin2 xsin2 x

= 1

b. cos2 x !1+ sin2 x = 0

! sin2 x + sin2 x = 0

6-125. a. f (x) = x2 3 b. g(x) = 2 f (x) ! 3 6-126.

2y + 2x = xy! y = 2xx"2 !!!!!6y " 6x = xy! y = 6x

6"x2xx"2 =

6x6"x

2x(6 " x) = 6x(x " 2)12x " 2x2 = 6x2 "12x

0 = 8x2 " 24x0 = 8x(x " 3)

x = 3!!!y = 2#33"2 = 6

6-127. See graph at right. a. y = 2x+7

x!7 x ≠ 7

b. 1. limx!7+

2x+7x-7 = " 2. lim

x!7"2x+7x-7 = "#

3. limx!"

2x+7x-7 = 2 4. lim

x!"#2x+7x-7 = 2


6-128. 3

d = h!1.62h

3h = d(h !1.62)

d = 3hh!1.62

6-129. a. sin 2! = 0

2 sin! cos! = 0sin! = 0 cos! = 0

! = 0, "2 , " , 3"

2 , 2"

b. sin2 ! " cos2 ! = 0(sin! " cos!)(sin! + cos!) = 0sin! " cos! = 0 sin! + cos! = 0sin! = cos! sin! = " cos!

! = #4 , 3#

4 , 5#4 , 7#

4

6-130. sin x = ! 3

7 , cos x = ! 2 107

sin 2x = 2 sin x cos x

2 sin x cos x = 2 " ! 37 " !

2 107 = 12 10

49

cos 2x = 2 cos2 x !1

2 ! 2 107( )2 !1 = 2 40

49( ) !1 = 8049 !

4949 =

3149

6-131. a. Exponential is reasonable if it really grows faster and faster. Linear fits well for this data

but it does not fit her hypothesis.

b. y = 12 1512( )x , with x = number of days since Monday.

c. y = 12 1512( )1 = 15

y = 12 1512( )4 = 12 ! 50625

20736( ) = 29.3

Perfect on Monday and Tuesday; 29.3 instead of 29 on Friday. It fits quite well.

d. 100 = 12 1512( )x

253 = 15

12( )xln 25

3( ) = x ln 1512( )

x = 9.502

The following Wednesday night or Thursday early morning. y = 100 when x = 9.502.


Lesson 6.4.1 6-132. b. Since cosine starts at a peak, we will not have to incorporate a horizontal shift. 6-133. The period stays consistent regardless of the oscillations. 6-134. Half of the period. 6-138. No, the height of the oscillations will decrease with time. 6-139. Only the amplitude is affected. We observed earlier that the period stays consistent. The

slinky will oscillate up and down until it comes to rest in the middle position. 6-140. The graph is approaching the vertical shift. Review and Preview 6.4.1 6-141. Amplitude 5!22 = 3

2 Vertical shift 2 + 1.5 = 3.5

Horizontal shift is right 2 units Period 4 = 2!b !!"!!b =

!2

y = 1.5 cos !2 (x " 2)( ) + 3.5 or

y = !1.5 cos "2 x( ) + 3.5 with a vertical flip instead of a horizontal shift

6-142. cos x = ! 5

3 , tan x = 25

, csc x = ! 32 ,!sec x = ! 3

5, cot x = 5

2

6-143. sin!1(x) :! ! "

2 ,"2#$ %& , cos!1(x) :! 0,"[ ] , tan!1(x) :! ! "

2 ,"2( )

6-144. tan!1 x is inverse tangent while cot x = 1

tan x . 6-145.

8m2

2m= 6!!!!! 4m = 6!!!!!4m = 36!!!!!m = 9


6-146. a. 2! x + 2y = 200

2y = 200 " 2! xy = 100 " ! x

b. A(x) = 2x(100 ! " x)A(x) = 200x ! 2" x2

6-147. Draw a line through B parallel to CD meeting AC at E. Then

AE = 60 cm , AB = 100 cm, and ABE is a right triangle. Hence BE = CD = 80. Let θ be the central ∠BAC. Then cos θ = 0.6, so ! " 0.927 radians. Thus the wire length around the large log is 80(2π – 2(0.927)) = 354.287 cm. The wire around the small log is 20(2(0.927)) = 37.092 cm in length and the wire between the logs is 2(80) cm. Thus, the total length is 354.287 + 37.092 +160 = 551.379 .

6-148. a. 5·3(x+2) != k·3x

3(x+2)

3x= k

5

3x+2!x = k5

9 = k5

k = 45!

b. 6·2(x+k) != 24·2x

2(x+k)

2x= 4

2x+k!x = 42k = 22

k = 2

Lesson 6.4.2 6-149. a. y = k b. amplitude (a) c. The high points are decreasing while the low points are increasing. 6-150. The data looks surprisingly linear in the ZoomStat window. 6-151. a. slope = 26.553!26.746

2!1 = !0.193y ! 26.746 = !0.193(x !1)

y = !0.193(x !1) + 26.746y = !0.193x + 0.193+ 26.746y = 26.939 ! 0.193x

b. y = !0.193(9) + 26.939y = !1.737 + 26.939 = 25.202

y = !0.193(10) + 26.939y = !1.93+ 26.939 = 25.009

CD

A B

E 20

20

60

80 + 20 = 100 θ


6-152. a. It is half way between them. b. 26.746 = 22.175 + a !m1

am = 4.571

a = 4.571m

26.553 = 22.175 + a !m2

4.378 = am2

a = 4.378m2

4.571m ! 4.378

m2= 0

4.571m ! 4.378 = 04.571m = 4.378m = 0.95778

a = 4.3780.957782

= 4.772

c. y = 22.175 + (4.772) !0.95778 p

y = 22.175 + (4.772) !0.957789

y = 22.175 + 3.237 = 25.412 y = 22.175 + (4.772) !0.9577810

y = 22.175 + 3.10 = 25.275

6-153. Exponential decay is better. 6-154. a. The exponential function approaches the resting position of the spring. b. 1.7 seconds c. p = x

1.7

d. y = 4.772(0.95778)x1.7

y = 4.772 (0.95778)11.7!

"#$%&x

y = 4.772(0.97494)x

e. y = 4.772(0.95778)x 1.7 cos 2!1.7 x( ) + 22.175

f. 22.175. Students should say that the spring approaches the model’s vertical shift.


Review and Preview 6.4.2 6-155. a. sin2 x

sin x(1+cos x) +(1+cos x)(1+cos x)sin x(1+cos x) = sin2 x+1+2 cos x+cos2 x

sin x(1+cos x) = 2+2 cos xsin x(1+cos x)

= 2(1+cos x)sin x(1+cos x) =

2sin x = 2 csc x

b. cos! (1"sin! )1"sin2 !

+ cos! (1+sin! )1"sin2 !

= cos!"cos! sin!+cos!+cos! sin!1"sin2 !

= 2 cos!cos2 !

= 2cos! = 2 sec!

6-156. a. 1! sin2 " = 0

sin2 " = 1sin" = ±1

" = #2 + 2#n,

3#2 + 2#n

b. 4 cos2 ! = 3

cos2 ! = 34

cos! = ± 34 = ± 3

2

! = "6 ,

5"6 ,

7"6 ,

11"6 , all + 2"n

6-157. a. 2x2 ! 2x ! 5 = 0

x = !(!2)± (!2)2 !4(2)(!5)2(2)

x = 2± 444 = 2±2 11

4 = 1± 112

b. 6x4 ! x2 ! 5 = 0

(6x2 + 5)(x2 !1) = 0

x2 !1 = 0!!or!!6x2 + 5 = 0

x2 = 1 !or!!!!!x2 " ! 56

x = ±1

c. x + 21 = 7 ! x

x + 21( )2 = 7 ! x( )2x + 21 = 49 !14 x + x

28 = 14 x

(28)2 = (14 x )2

784 = 196xx = 4

d. 2 + 5x!1 =

12(x!1)2

2(x !1)2 + 5(x !1) = 122x2 ! 4x + 2 + 5x ! 5 = 12

2x2 + x !15 = 0(2x ! 5)(x + 3) = 0

x = !3, 52

6-158. x2 + (x + 4)2 = (x + 8)2

x2 + x2 + 8x +16 = x2 +16x + 64x2 ! 8x ! 48 = 0

(x !12)(x + 4) = 0x = 12 (since x " !4)

The lengths of the sides of the triangle are 12, 16, and 20.


6-159. y = !7 cos "

0.7 (x ! 2.3)( ) +15

Amplitude: 22!82 =142 =7 Period: 1.4 = 2!b !!"!!b =

!0.7

Horizontal shift: 2.3 (seconds) to the right Vertical shift: 15 (inches) up

12 = !7 cos "0.7 (x ! 2.3)( ) +15

0.4286 = cos "0.7 (x ! 2.3)( )

1.1279 = "0.7 (x ! 2.3)

0.2513 = x ! 2.3x = 2.552

x = 0.649, 1.152, 2.049, 2.552, 3.449, 3.952, 4.849

6-160. 2 sin! cos! = cos! cos" + sin! sin "

2 sin! cos! = # cos!2 sin! cos! + cos! = 0cos!(2 sin! +1) = 0

2 sin! +1 = 0 or cos! = 0

sin! = # 12

! = "2 + "n, "6 + 2"n, 7"

6 + 2"n 6-161. a. y = k

x+6

1 = k1+6

k = 7

y = f (x) = 7x+6

b. f (!3) = 7!3+6 =

73

f (0) = 70+6 =

76

f 13( ) = 7

1 3+6 =719 3 = 7 "

319 =

2119

f 1a( ) = 7

1 a+6 =7

1 a+6a a =7

1+6a a

= 7 " a1+6a =

7a1+6a


Closure Chapter 6 CL 6-162. a. b. c. !

6 , the function sin!1(x) can only have one output. CL 6-163. a. 2 cos x = !1

cos x = ! 12

x = 2"3 ,

4"3

b. sin2 x = 34

sin x = ± 32

x = !3 ,

2!3

c. tan x = !1

x = 3"4 ,

7"4

CL 6-164. a. y = 2 sin(2x) + 2, y = 2 cos 2 x ! "

4( )( ) + 2

b. y = !3sin 12 x ! "

4( )( ) !1, y = 3 cos 12 x + "

4( )( ) !1 CL 6-165. a. See triangles at right. b.

6sin 34!

= 8sin C

8 sin 34! = 6 sinC4.47356 = sinC

0.7456 = sinC!C = 48.21 or !C = 131.79!

!B = 180! " 34! " 48.21! = 97.79!

!B = 180! "131.79! " 34! = 14.21!

6sin 34!

= ACsin 97.79!

6 ! sin 97.79! = AC ! sin 34!

5.9446 = 0.5592ACAC = 10.63 cm

6sin 34!

= ACsin14.21!

6 ! sin14.21! = AC ! sin 34!

1.4729 = 0.5592ACAC = 2.63 cm

c. If !B = 97.79º : If !B = 14.21º : A = 1

2 (6)(8) sin(97.79º ) = 23.78cm2 A = 1

2 (6)(8) sin(14.21º ) = 5.89cm2

Difference = 23.78 – 5.89 = 17.89 cm2

f(x)=sin(x)

!/2 ! 3!/2 2!

-1

-0.5

0.5

1

C

6 cm

A

B

34°

8 cm

A C

B

6 cm 34°

8 cm

π/6 5π/6

12

12


CL 6-166. a.

tan!1 2

3( ) = 33.7! b. tan!1(!2) = !63.4!

c. 180! ! 63.4! ! 33.7! = 82.9! CL 6-167. sin!1(x) : ! "

2 , "2#$ %& , cos!1(x) : 0,"[ ] , tan!1(x) : ! "2 , "2( )

CL 6-168.

a. cos A = 12

13 b. sin B = 35

c. cos(A + B) = cos A cos B ! sin A sin B

cos(A + B) = 1213 "

45 +

513 "

35 =

4865 !

1565 =

3365

CL 6-169. a. 10 !2 sin 2x cos 2x = 10 sin(2 !2x) = 10 sin(4x) b. sin(! " x) = sin ! cos x " sin x cos!

= 0 # cos x " ("1) # sin x= sin x

c. ! cos2 " + sin2 " = !(cos2 " ! sin2 ")= ! cos(2")

d. cos(x + !2 ) = cos x cos

!2 " sin x sin

!2

= 0 #cos x " (1) #sin x= " sin x

CL 6-170. a. 2(1! sin2 x) + sin x = 2

2 ! 2 sin2 x + sin x = 22 sin2 x ! sin x = 0sin x(2 sin x !1) = 0

sin x = 0 or 12

x = 0, " , 2" , "6 ,

5"6

b. 2(1! sin2 x) + sin x = 22 ! 2 sin2 x + sin x = 22 sin2 x ! sin x = 0sin x(2 sin x !1) = 0

sin x = 0 or 12

x = 0, " , 2" , "6 ,

5"6 , all + 2"n

Solution continues on next page. →


CL 6-170. Solution continued from previous page. c. sin x ! sin(2x) = 0

sin x ! 2 sin x cos x = 0sin x(1! 2 cos x) = 0

sin x = 0 cos x = 12

x = 0, " , 2" , "3 ,

5"3

d. sin x ! sin(2x) = 0sin x ! 2 sin x cos x = 0sin x(1! 2 cos x) = 0

sin x = 0 cos x = 12

x = 0, " , 2" , "3 ,

5"3 , all + 2"n

CL 6-171. Amplitude 6!1.52 = 4.5

2 = 2.25 Period 2.5 = 2!b !!"!!b =

!1.25 = 0.8!

Horizontal shift 2.5 seconds to the right Vertical shift 3.75 feet up h = 2.25 sin 0.8! (t " 3.125)( ) + 3.75 a. 3.055 feet b. At 0.2704 and 2.5 – 0.2704 = 2.2296 seconds CL 6-172. a. 18(1.03)x ! 20 = 300

18(1.03)x = 320

(1.03)x = 17.7778

log1.03 1.03x = log1.03 17.7778

x = log 17.7778log 1.03 = 97.364

b. log2 5x( ) = 32log2 5x( ) = 23

5x = 8

x = 85

c. x2.7 = 1608 = 20

(x2.7 )1 2.7 = 201 2.7

x = 3.033

d. log3 x+5x!1( ) = 1

3log3 x+5

x!1( ) = 31x+5x!1 = 3

3x ! 3 = x + 52x = 8x = 4

CL 6-173. (!1)3 ! 4 = (!1)a + b

!5 = !a + b 12 ! 8 = a + b

!7 = a + b !7 = a + b

!5 = !a + b!12 = 2b

b = !6

!7 = a ! 6a = !1

y = !1x ! 6

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Chapter 6: Extending Periodic Functions - Weeblyachsprecalc.weebly.com/uploads/1/9/8/2/19823981/cpm_precalculus... · Chapter 6: Extending Periodic Functions Lesson 6.1.1 6-1. a