Chapter 6: Extending Periodic Functions -...
Transcript of Chapter 6: Extending Periodic Functions -...
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).
CPM Educational Program © 2012 Chapter 6: Page 2 Pre-Calculus with Trigonometry
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
CPM Educational Program © 2012 Chapter 6: Page 3 Pre-Calculus with Trigonometry
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"# $% = 7 +132 + 5
6-14.
x = b ! y3(b ! y) ! 2y = a3b ! 3y ! 2y = a
!5y = a ! 3b
y = 3b ! a5
x = b ! 3b ! a5
x = 5b5
! 3b ! a5
= a + 2b5
6-15. 4
5 =x32
128 = 5x1285 = x
x = 25.6 feet
6-16. a. 9!
0.2 =36!x
9! " x = 36! "0.29! x = 7.2!
x = 7.29 = 0.8 liters/hr
b.
6-17. g(!1) = (!1)2 ! 2(!1)
g(!1) = 1+ 2 = 3
g(3) = 32 ! 2(3)g(3) = 9 ! 6 = 3
g(a) = a2 ! 2(a)g(a) = a2 ! 2a
g(t ! 2) = (t ! 2)2 ! 2(t ! 2)g(t ! 2) = t2 ! 4t + 4 ! 2t + 4
= t2 ! 6t + 8
V = 13 ! "62 "10 = 120!
12 "V = 1
2 "120! = 60!
60! = 13 ! r
2 "h
180 = r2 53 r( )
108 = r3
r = 4.76229!0.2 =
4.76222!x
9! " x = 22.679! "0.29! x = 4.536!
x = 4.5369 = 0.504 liters/hr
CPM Educational Program © 2012 Chapter 6: Page 4 Pre-Calculus with Trigonometry
6-18. a. x2
x2 !x! 1x2 !x
= 3
x2 !1x(x!1) = 3
(x+1)(x!1)x(x!1) = 3
x+1x = 33x = x +12x = 1
x = 12
b. xx2 !25
+ x!5x2 !25
= 1
2x!5x2 !25
= 1
2x ! 5 = x2 ! 250 = x2 ! 2x ! 200 = x2 ! 2x +1!1! 2021 = (x !1)2
± 21 = x !1
x = 1± 21
Lesson 6.1.2 6-19. b. Inverses are symmetric about the line y = x . c. No, because it does not pass the vertical line test. 6-20. a. ! "
2 ,"2#$ %&
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,![ ] .
CPM Educational Program © 2012 Chapter 6: Page 5 Pre-Calculus with Trigonometry
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
CPM Educational Program © 2012 Chapter 6: Page 6 Pre-Calculus with Trigonometry
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
CPM Educational Program © 2012 Chapter 6: Page 7 Pre-Calculus with Trigonometry
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
CPM Educational Program © 2012 Chapter 6: Page 8 Pre-Calculus with Trigonometry
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.
CPM Educational Program © 2012 Chapter 6: Page 9 Pre-Calculus with Trigonometry
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#$%
&%
h(x) =!2(x ! 3) +1 !!for x < !2
2(x ! 3) !1 ! 2 for x " !2#$%
&%
CPM Educational Program © 2012 Chapter 6: Page 10 Pre-Calculus with Trigonometry
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
CPM Educational Program © 2012 Chapter 6: Page 11 Pre-Calculus with Trigonometry
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
CPM Educational Program © 2012 Chapter 6: Page 12 Pre-Calculus with Trigonometry
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
CPM Educational Program © 2012 Chapter 6: Page 13 Pre-Calculus with Trigonometry
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
CPM Educational Program © 2012 Chapter 6: Page 14 Pre-Calculus with Trigonometry
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
CPM Educational Program © 2012 Chapter 6: Page 15 Pre-Calculus with Trigonometry
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)
CPM Educational Program © 2012 Chapter 6: Page 16 Pre-Calculus with Trigonometry
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
CPM Educational Program © 2012 Chapter 6: Page 17 Pre-Calculus with Trigonometry
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
CPM Educational Program © 2012 Chapter 6: Page 18 Pre-Calculus with Trigonometry
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
CPM Educational Program © 2012 Chapter 6: Page 19 Pre-Calculus with Trigonometry
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
CPM Educational Program © 2012 Chapter 6: Page 20 Pre-Calculus with Trigonometry
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
CPM Educational Program © 2012 Chapter 6: Page 21 Pre-Calculus with Trigonometry
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
CPM Educational Program © 2012 Chapter 6: Page 22 Pre-Calculus with Trigonometry
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
CPM Educational Program © 2012 Chapter 6: Page 23 Pre-Calculus with Trigonometry
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
CPM Educational Program © 2012 Chapter 6: Page 24 Pre-Calculus with Trigonometry
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
CPM Educational Program © 2012 Chapter 6: Page 25 Pre-Calculus with Trigonometry
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
CPM Educational Program © 2012 Chapter 6: Page 26 Pre-Calculus with Trigonometry
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
CPM Educational Program © 2012 Chapter 6: Page 27 Pre-Calculus with Trigonometry
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.
CPM Educational Program © 2012 Chapter 6: Page 28 Pre-Calculus with Trigonometry
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
CPM Educational Program © 2012 Chapter 6: Page 29 Pre-Calculus with Trigonometry
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
CPM Educational Program © 2012 Chapter 6: Page 30 Pre-Calculus with Trigonometry
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.
CPM Educational Program © 2012 Chapter 6: Page 31 Pre-Calculus with Trigonometry
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.
CPM Educational Program © 2012 Chapter 6: Page 32 Pre-Calculus with Trigonometry
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
CPM Educational Program © 2012 Chapter 6: Page 33 Pre-Calculus with Trigonometry
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
CPM Educational Program © 2012 Chapter 6: Page 34 Pre-Calculus with Trigonometry
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. →
CPM Educational Program © 2012 Chapter 6: Page 35 Pre-Calculus with Trigonometry
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