Chapter 6: Etending Periodic Functions Lesson 6.. 6-. a. The graphs of y = sin and y = 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 = 0.5. d. = 4! and = 5!. Students may use unit circle or the graph. 6-. Draw a vertical line at =. The angles that satisfy the equation are =! and = 5!. 6-. 6-4. A horizontal line drawn at y = does not intersect the unit circle. The value is not in the range for y = sin. Eamples of trig equations: cos =, csc = 0 Eamples of non-trig equations: + + 4 = 0, + 4 =! + 6-5. a. sin + = 0 sin =! = " c. cos =! = " 4, 5" 4 b. cos =! cos =! = ", 4" d. sin! = 0 sin = sin = = ", " 6-6. a. All real numbers. b.! " y " c. The functions both have a period of!, so a shift of that size would not affect either function. 6-7. a. There would be an infinite number of solutions. b. solutions: 0 and π c. Infinitely many. d. An integer multiple of!, because it is the period (!n for n an integer). CPM Educational Program 0 Chapter 6: Page Pre-Calculus with Trigonometry
6-8. a. There are an infinite number of solutions. There are solutions on the graph given. 5! b. 6 +! = 5! 6 +! 6 = 7! 6 5! 6 "! = 5! 6 "! 6 = " 7! 6 c. Add!n to! 6, 5!, n is any integer. 6! d. 6 + 4! =! 6 + 4! = 5! 6 6 5! 6 + 4! = 5! 6 + 4! = 9! 6 6 6-9. a. The y-coordinates of the points are. e.! " 6! 5" =! " 6! 0" =! " 6 6! 5" 6! 5" =! 5" 6! 0" =! 5" 6 6 b. Answers vary, but going around the circle! would take us back to the same place as! 6 or! 5! 6. 6-0. a. sin! = sin = sin = b.! +!n,! +!n = ", " Review and Preview 6.. 6-. a. Since the string is 0 inches in length, the maimum point will be 0 inches above the minimum. b. 0 = 5 c. 5 + 5 = 0 d.!.5 =! 5 =! " 5 = 4! 5 ( ) + 0 f. h =!5 cos 4" 5 (t) 6-. a. csc 5! 6 = sin 5! 6 = = " = b. tan! = sin! cos! / = 0 " undefined c. cot 5! = cos5! sin 5! = = " # = = " " " d. sec 7! 6 = sin 7! 6 = = # = = " " " " e.! cos CPM Educational Program 0 Chapter 6: Page Pre-Calculus with Trigonometry
6-. a. 0 = = = b. "! (k! ) + 7 left-sum = " #! (! ) + 7 $ % + " #! (! ) + 7 $ % + " #! (! ) + 7 $ % = 7 + + 5 k= 6-4. = b! y (b! y)! y = a b! y! y = a!5y = a! b y = b! a 5 = b! b! a 5 = 5b 5! b! a = a + b 5 5 6-5. 4 5 = 8 = 5 8 5 = = 5.6 feet 6-6. 9! a. 0. = 6! 9! " = 6! " 0. 6-7. 9! = 7.! = 7. 9 = 0.8 liters/hr g(!) = (!)! (!) g(!) = + = g() =! () g() = 9! 6 = g(a) = a! (a) g(a) = a! a g(t! ) = (t! )! (t! ) g(t! ) = t! 4t + 4! t + 4 = t! 6t + 8 b. V =! " 6 "0 = 0! "V = "0! = 60! 60! =!r " h 80 = r 5 r 08 = r ( ) r = 4.76 9! 0. = 4.76! 9! " =.679! " 0. 9! = 4.56! = 4.56 9 = 0.504 liters/hr CPM Educational Program 0 Chapter 6: Page Pre-Calculus with Trigonometry
6-8. a.!!! =! (!) = (+)(!) (!) = + = = + = = b.!5 +!5!5 =!5!5 =! 5 =! 5 0 =!! 0 0 =! +!! 0 = (! ) ± =! = ± Lesson 6.. 6-9. b. Inverses are symmetric about the line y =. c. No, because it does not pass the vertical line test. 6-0. a. #! ", " % $ & b. The domain of y = sin! will be the range of y = sin, so the domain is!, [ ]. 6-. a.! =.047 b. It is not in the range of y = sin!. The inverse of sine only selects one of the infinitely many solutions to the equation. c. =! +!n or! +!n d. You have to use the unit circle or a wave. 6-. a. It does not pass the vertical line test. b. 0,! [ ] c. The domain of y = cos! is the range of y = cos, which is!, y = cos! is [ 0,! ]. [ ]. The range of CPM Educational Program 0 Chapter 6: Page 4 Pre-Calculus with Trigonometry
6-.! y! y!! " y = sin! () :!!D :[!,],!R : #! ", " % $ & y = cos! () :!!D :[!,],!R :[ 0, " ] 6-4. a. 0.05 b.! " 0.05 =.87 c. 0.05 +!n,.87 +!n, for n an integer. 6-5. a. vertical line b..66 c. 5.07 =! ".66 d..66 +!n, 5.077 +!n or ±.66+!n, n an integer Review and Preview 6.. 6-6. a. It is not in the range of y = cos!. cos! selects only one of the infinitely many solutions to the equation. b. =! +!n or 5! +!n c. You have to draw and think. 6-7. tan = sin cos = 0! sin = 0 = ", ", ", 4" = n", n is any integer 6-8. a. The equation cos =!0. 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 π, and include the negative values. = ±.875 +!n, where n is an integer. CPM Educational Program 0 Chapter 6: Page 5 Pre-Calculus with Trigonometry
6-9. See diagram at right. a.! "! b. 4! c. 4! d. 6 6-0. + = = 5 = ± 5 cos! = " 5 6-. a. 6 = 0 + 8! (8)(0) cos 6-. 6 = 64! 60 cos!8 =!60 cos 0.8 = cos cos! 0.8 = cos! (cos ) a. log 64 = 6.9! ( ) = log ( 64! ) = log ( 6 )! = log!6 =!6 b. sin 60! = 8 sin 70! sin 70! = 8 sin 60! 0.997 = 4.587 b. log 8 = 0 = 5.8 c. log 8 8 = d. log (64) = log ( 6 ) = 6 e. impossible f. log 5 5 ( ) ( ) = log 5 ( 5 ) ( ) = = log 5 5 6-. a. y!4 y +y y!y = y (!+) y (!) = (!)(!) y(!) = (!) y b. (+h)! = +h+h! h h = h+h h = h(+h) = + h h CPM Educational Program 0 Chapter 6: Page 6 Pre-Calculus with Trigonometry
6-4. a.! f () + b. f (!) Flipped over -ais and up. Flipped over y-ais and stretched vertically. y y c. f () Asymptotes at =!, 0, and. y Lesson 6.. 6-5. The Law of Sines calculation results in the sine of the angle at Icy s being greater than. The Law of Cosines calculation yields a quadratic equation with no real solutions. 6-6. a. 0 sin 8! = 0 sin I 0! 0.4695 = 0 sin I 4.085 0 = sin I sin " 0.7045 = sin " sin I 44.8! = #I (or!i = 5.!, but don ot point this out yet) b.!d = 80! " 8! " 44.8! = 07.! 0 sin 8! = d sin 07. d # 0.4695 = 0 sin 07.! d = 9.056 0.4695 d = 40.69 m c. Katya missed the possibility that!i could be obtuse.!i = 80! " 44.8! = 5.!!D = 80! " 5.! " 8! = 6.8! 0 sin 8! = d sin 6.8! d # 0.4695 = 0 sin6.8! d = 5.7806 0.4695 d =. m CPM Educational Program 0 Chapter 6: Page 7 Pre-Calculus with Trigonometry
6-7. 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 nd quadrant. 6-8. a. 0 sin 90! = a sin 0!!0 = a! a = 5 c. 0 sin C = sin 0! 0!sin 0! =!sin C 5 = sin C 5 " sin C Not possible since the range of sine is [#, ]. b. 0 sin C = 5 sin 0! 0! sin 0! = 5! sin C 5 = 5 sin C = sin C "C = 90! 0 d. sin C = 7 sin 0! 0!sin 0! = 7!sin C 5 = 7 sin C 5 7 = sin C ( ) "C = sin # 5 7 "C = 45.58! or "C = 80! # 45.6! = 4.4! e.!acb = 80! "!BC C # = 80! "!B C# C since!bc C " is isosceles. f. Supplementary angles have the same sine. g. One triangle. 6-9. 0 triangles if a < c sin A ; triangle if a = c sin A or a! c, triangles if c sin A < a < c. Review and Preview 6.. 6-40. 9 sin 4! = 8 sin C 8! sin 4! = 9! sin C 4.47 = 9 sin C 4.47 9 = sin C "C = sin "C = 9.8! ( ) # 4.47 9!B = 80! " 4! " 9.8! = 6.! 9 sin 4! = AC sin 6.! AC sin 4! = 9 #sin6.! AC = 8.075 = 4.44 cm 0.559 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 0 Chapter 6: Page 8 Pre-Calculus with Trigonometry
6-4. a. sin = 4 5 sin! 4 5 ( ) = 0.97 c. 0.97 + pn,.4 + pn, n is an integer. b. = 0.97 and! " 0.97 =.4 6-4. g() = k. = k 4 k = 6!. = 9. g(6) = 9. 6 g(6) = 96 5! 6 = 8 5! = 8 5 g(!) = 9. (!) g(!) = 96 5 " 9 = 5 " = 5 6-4. y = + = + + y y+ = y++y y+ (y + ) = y + y + = y + y! y =! y(! ) =! y =!! =!! f! () =!! 6-44. 6-45. g() = (+)(! ) Asymptotes occur when the denominator equals zero. This occurs when = 0,!,. +cos! ("cos!)(+cos!) + "cos! ("cos!)(+cos!) +cos! +"cos! = "cos =! sin =! csc! 6-46. 6-47. f () = 7(9)! = ( )! =! = +! = + = () #%!(! ) +! for <! f (! )! = $ &% (! )!! for "! #%!(! ) +!!for <! h() = $ &% (! )!! for "! CPM Educational Program 0 Chapter 6: Page 9 Pre-Calculus with Trigonometry
Lesson 6..4 6-48. a. You would find vertical asymptotes when cos = 0. These occur at =! ",! ", ", ". b. This would be when the graph of tan crosses the -ais, which are the roots, and they occur at =!, "!, 0,!,!. 6-49. a.! n", where n is any odd integer. b. All real numbers. c. y = 0, = +!n, n is any integer. d. = n!, where n is any odd integer. 6-50. a. Restrict the range. b. Range: # $! ", " % & 6-5. a. lim!" tan # () = $ b. lim!"# tan" () = " $ 6-5. tan! = opposite adjacent = y 6-5. 6-54. 6-55. tan! = ( ) tan " tan! = tan "! = 6.6! or 0.464 radians adjacent side = 45 =.5 tan! = opposite adjacent = 8.5 ( ) tan " tan! = tan " 8.5! = 9.57!! =. radians tan. =.57 approimate slope =.57 CPM Educational Program 0 Chapter 6: Page 0 Pre-Calculus with Trigonometry
Review and Preview 6..4 6-56. a. sin! = 0 sin = sin = c.! sin = 0 = " 6 + "n, 5" 6! cos =! cos = = " 4 + "n, " 4 + "n, n is an integer + "n, n is an integer b. + cos = 0 cos =! d. cos +.8 = 0 cos =! =! = " + "n, n is an integer cos =!.8 cos /> " no solution 6-57. Yes, the first is the inverse function, the second the reciprocal function of y = cos. 6-58. 6-59. sin = 0. has infinite solutions unless we are working with a restricted values of. The epression sin! 0. = has only one solution when sin! is a function. It is false. For eample, take a =! 6, b =!. sin! +! 6 ( ) = sin (! 6 +! 6 ) = sin (! ) = ( ) + sin (! 6 ) = + = + but sin! " 6-60.!+ 8 + a = ( +b + b )!!+ 8 + a = +4b + b 8 = 4b b = a =!b =! = 8 6-6. 6-6. Amp. =, horizontal shift = to the right, vertical shift = up, period =!! =! "! = 4. tan! = sin! cos! = " = # " = " CPM Educational Program 0 Chapter 6: Page Pre-Calculus with Trigonometry
6-6. a. slope of PR =!6 4!(!4) =!4 8 =! 9 perpendicular slope = 9 midpoint of PR =!4+0 ( ) = (5, 4), 6+ y! 4 = 9 (! 5) y = 9 (! 5) + 4 b. slope of median =!4!5 = 8! =! 8 y! 4 =! 8 (! 5) y =! 8 (! 5) + 4 c. slope of PR =!6 4!(!4) =!4 8 =! 9 6-64. perpendicular slope = 9 y! = 9 (! ) y = 9 (! ) + 0 =.5, =.75, =.5, =.75,! 4 =.5, 5 =.75, 6 = 4.5, 7 = 4.75 k = 0.5k +.5 sum = 7 0.5k+.5 k=0! ".600 Lesson 6.. 6-65. Laurel is. Hardy s equation only shifts the graph! 6 H () = sin! " ( ). ( ) = sin (! " 6 ) to the right since 6-66. a. =! b. =! 6 c. H () = sin! " ( ) ( ) = sin (! " 6 ) 6-67. y = sin((! " )) + 4 6-68.!! (!5) a. Amplitude = = Horizontal shift is! to the right. Vertical shift is down. The period is! =!. ( ( ))! b. y = sin! " CPM Educational Program 0 Chapter 6: Page Pre-Calculus with Trigonometry
6-69. ( ( )) a. y = cos(!( + )) " b. y = sin! " 6-7. a. (0.4, 46) and (., 6) b. Period = (.! 0.4) =.6, Amplitude = Vertical shift = 6 + 0 = 6 + 0 = 6. c. One possible answer is h(t) = 0 cos! 6! 46 = 0 ((.6 ) (t " 0.4) ) + 6. = 0, horizontal shift 0.4 or.4, Review and Preview 6.. 6-7. y = sin! ( " ) 6-7. 5 + (leg b) = 8 ( ) + (leg b) = 64! 5 4 4 y 4 leg b = 9 a. sin! = 5 8 b. cos! = " 9 c. tan! = 8 5 8 = 5 " 9 8 8 # " 8 = " 5 # 9 = " 5 9 9 9 9 9 6-74. a. The range of sine and cosine is! " y ". b. A fraction can equal without the numerator being and the denominator being 7. For 7 eample, 0. 0.7 = 7. c. tan! tan = tan! ( 7 ) = 0.405 or! 0.405 + " =.546 CPM Educational Program 0 Chapter 6: Page Pre-Calculus with Trigonometry
6-75. a.! 4! = 0 ( + )(! 7) = 0 =!, 7 c. + = 0 +! 0 = 0 (! 5)( + ) = 0 = 5,! b. (! )( + ) = 4!! = 4!! 6 = 0 (! )( + ) = 0 =!, d. 6 + 5 = 5 6 + 5! 5 = 0 (! 5)( + 5) = 0 = 5,! 5 6-76. 6-77. 6-78. tan!csc sec = tan 8! = 0.5 sin cos! sin cos = cos cos y! 9 = ±0.5(! 85) sec!tan sin = cos! sin cos = sin sin cos sin = = sin cos! sin = cos = sec 6-79. ( ) " a. y =! cos()! b. y = sin +! 4 c. y = sec() d. y = tan! 6-80. h = kv r 5 = k0 4 60 = 0k k = h = V r h =!0 9 h = 0 9 = 0 CPM Educational Program 0 Chapter 6: Page 4 Pre-Calculus with Trigonometry
Lesson 6.. 6-8. a. cos! " 4 b. cos! " 4 ( ) = cos sin ( " 4 ) + sin cos ( " 4 ) ( ) = cos # + sin # ( ) = (cos + sin ) ( ) = cos + sin c. cos! " 4 d. cos! " 4 6-8. a. cos(90! -!) = cos 90! cos! + sin 90! sin! = 0 " cos! + " sin! = sin! b. sin(90! -!) = sin 90! cos! " cos 90! sin! = # cos! + 0 #sin! = cos! c. cot! = cos! sin(90º "!) = = tan(90º "!) sin! cos(90º "!) d. csc! = sin! = cos(90! "!) = sec(90! "!) 6-8. a. cos! = " 5 b. sin! = " 7 4 ( ) = " c. sin(! " #) = 4 5 $ ( " 4 ) " (" 5 ) $ " 7 4 d. cos(! + ") = # 5 $ ( # 4 ) # ( 4 5 ) $ # 7 4 = # 9 0 + 4 7 0 = 9+4 7 0 0 " 7 0 = "" 7 0 Review and Preview 6.. 6-84. a. 0 b. -coordinate: 5!. = B! 5!!!"!!B = 0!. = 8.69 (8.69, 5) c. -coordinate:.! 5 = 5! C!!!"!!C = 0!. =!. (., 5) CPM Educational Program 0 Chapter 6: Page 5 Pre-Calculus with Trigonometry
6-85. a. Amplitude is 0. Horizontal shift is 5 to the right. Vertical shift is 4 up. The period is!! =! "! = 4. b. See graph at right. 6-86. a. 0 sin! " 5 ( ( )) + 4 = 0 If u =! ( " 5) 0 sin u = "4 sin u = " 5 ( ) sin " sin u = sin " " 5 u =!0.45 " (! 5) =!0.45! 5 =!.6 = 4.78! 4 = 0.78 b. y = 0 sin (! " 5 ( )) + 4 =.6 6-87. cos(! -") = cos! cos" + sin! sin" = #cos" + 0 $sin " = # cos" 6-88. sin(! -") = sin! cos" # cos! sin" = 0 $ cos" + #(#) $ sin " = sin " 6-89. a.!,! b. sin + = 0 sin =! sin =! = 5" 4, 7" 4 c. cos ( sin + ) = 0 cos = 0 or sin + = 0 =!,!, 5! 4, 7! 4 d. cos ( sin + ) = 0 cos = 0 or sin + = 0 =! +!n, 5! 4 +!n, 7! 4 +!n 6-90. (csc + cot )(! cos ) = ( ) (! cos ) = sin + cos sin +cos ( sin ) (! cos ) =!cos = sin sin sin = sin 6-9. y! +! y y! +! = y " y! + y "! y y " y! + y "! = 4 + y 4 y + y CPM Educational Program 0 Chapter 6: Page 6 Pre-Calculus with Trigonometry
Lesson 6.. 6-9. y = 0 cos! ( " ) + 44 6 64! 4 Amplitude: = 40 = 0 inches Period: =! b!!"!!b =! 6 Horizontal shift: (hours) to the right Vertical shift: 44 (inches) up a. y = 0 cos! ("0.5 " ) + 44 6 y = 0 cos(".09) + 44 y = 5.764 + 44 y = 49.8 inches b. 7 tall = inches = 0 cos! ( " ) + 44 6 " 0 = cos! ( " ) 6 ( ) =! 6 cos " " 0 4.54 = " ( " ) = 6.54 # 6 hours minutes pm " 6 hours minutes # 9 : 9 a.m. 6-9.. h = 4 cos!(t ".5) + 4 Amplitude: 68! 0 = 4 inches Period: =! b!!"!!b =! Horizontal shift:.5 (seconds) to the right Vertical shift: 4 (centimeters) up a. h = 4 cos!(5.6 ".5) + 4 h = 4 cos(45.08) + 4 h = 5.457 + 4 h = 49.44 cm b. = 4 cos!( ".5) + 4 cos " " 4 " = cos!( ".5) 4 ( ) =!( ".5).745! = ".5 0.74 = ".5 =.974 sec. h = 4 cos (! ( - ) ) + 5 Amplitude: 9! = 4 feet Period: =! b!!"!!b =! Horizontal shift: (seconds) to the right Vertical shift: 5 (feet) up =.5 " 0.74 = 0.56 sec a. h = 4 cos! (5.4 " ) + 5 h = 4 cos(7.09) + 5 h = 4 # 0.669+ 5 h = 6.677 ft b. 7. = 4 cos! ( " ) + 5 0.55 = cos! ( " ) cos " (0.55) =! ( " ) 0.479 = " =.47 sec = " 0.479 =.58 sec CPM Educational Program 0 Chapter 6: Page 7 Pre-Calculus with Trigonometry
( ) + 54. d = 9 sin! (t " 5.5) 8! 5 Amplitude: = 9 cm Period: 6 =! b " b =! Horizontal shift: 5.5 (seconds) to the right Vertical shift: 54 (centimeters) up a. h = 9 sin! (8 " 5.5) + 54 h = 9 sin(.68) + 54 h = 4.5 + 54 h = 68.5 cm 4. A =. sin (! ( t.5) ) +.7.8! 0.6 Amplitude: =. liters Period: 6 =! b!!"!!b =! Horizontal shift:.5 (seconds) to the right Vertical shift:.7 (liters) up ( ) +.7 a. A =.sin! (.5 ".5) A =.sin(0) +.7 A =.7 liters ( ) + 8 CPM Educational Program 0 Chapter 6: Page 8 Pre-Calculus with Trigonometry b. b.. =.sin! (t ".5) +.7 0.5455 = sin! (t ".5) sin " (0.5455) =! (t ".5) 0.5509 = t ".5 t = 4.05 seconds 5. h = cos 8! ( " 0.5) 76! 0 Amplitude: = cm Period: 4 =! b!!"!!b =! # 4 = 8! Horizontal shift: 0.5 (seconds) to the right Vertical shift: 8 (cm) up a. h = cos ( 8! (5. " 0.5) ) + 8 b. 59 = cos ( 8! ( " 0.5) ) + 8 h = cos(4.56) + 8 59 = cos ( 8! ( " 0.5) ) + 8 h = # 0.045 + 8 cos " ( h = 40.404 cm ) # 8! = " 0.5 = 0.075 ( ) + 84 6. F = 9 sin! (t -0) 0! 65 Amplitude: = 9 degrees Period: 4 =! b!!"!!b =! Horizontal shift: 0 (hours) to the right Vertical shift: 84 (degrees) up ( ) + 84 ( ) + 84 a. F = 9 sin! (" 0) F = 9 sin! F = 4.98 + 84 F = 88.98! sin ( ) + 84 ( (t " 0) ) ( 9 ) =! (t " 0) b. 98 = 9 sin! (t " 0) 4 = 9 sin! sin = 9 sin! (t " 5.5) + 54 " 9 = sin! (t " 5.5) ( ) =! " " 9 " 4 (t " 5.5) "0.77 = t " 5.5 t = 4.767 seconds.64 = t " 0, t =.64.64 hours after noon or about :0 p.m.
7. h = 5.5 sin ( 5! (t ".4) ) +.5 Amplitude: 9! 8 = 5.5 cm Period: 4 5 =! 0!!!"!!b = b 4 = 5! Horizontal shift:.4 (seconds) to the right Vertical shift:.5 (centimeters) up a. h = 5.5 sin ( 5! (5 ".4) ) +.5 b. = 5.5 sin ( 5! (t ".4) ) +.5 h = 5.5 sin(0) +.5 " 0.5 5.5 = sin ( 5! (t ".4) ) h =.5 cm sin " (" 0.5 5.5 ) = 5! (t ".4) "0.0948 = t ".4 t =.05 Subtracting four periods from this ( 0.8! 4 =. ) gives.05!. = 0.05 seconds. 8. h = sin(p(t!.5)) + 7 Amplitude: 6! 0 = cm Period: =! b!!"!!b =! Horizontal shift:.5 (seconds) to the right Vertical shift: 7 (cm) up a. h = sin(!(0 ".5)) + 7 h = sin(5.86) + 7 h = + 7 = 0 cm ( ) +! 9. h = 6 cos (t " 5) 4 b. 5 = sin(!(t ".5)) + 7 " 9 = sin(!(t ".5)) ( ) =!(t ".5) sin " " 9 "0. = t ".5 t =.9 seconds Amplitude: 8! 6 = 6 cm Period: 8 =! b!!"!!b =! 4 Horizontal shift: 5 (seconds) to the right Vertical shift: (cm) up ( ) + a. h = 6 cos! (6 " 5) 4 h = 6 cos(6.494) + h = "4.46 + = 7.757 cm ( ) + ( (t " 5) ) b. 6 = 6 cos! (t " 5) 4 cos " 4 6 = cos! 4 ( ) =! 4 (t " 5).0709 = t " 5, t = 6.0709 seconds CPM Educational Program 0 Chapter 6: Page 9 Pre-Calculus with Trigonometry
Review and Preview 6.. 6-94. a. Amplitude is 4. b. Horizontal shift is!. 6-95. a. ( ) Vertical shift is. y = + 4 cos! " y = + 4 sin Other answers are possible. 5 b.! 5 4 c.! 4 + 5! " 5 = 4 + " 5 = "5 5 d.! 4 +! " 5 = 5 4 + " 5 e. 0.459 = 5 " 5 6-96. sin! = " 5 = " 5 tan! = " "4 = 4 csc! = 5 " = " 5 sec! = 5 "4 = " 5 4 cot! = "4 " = 4 6-97. a.! b.!! = c.!! 5 =! " 5! = 0 d.!! 5 =! " 5! = 0 6-98. 6-99. cos (! + " ) = cos! cos " # sin! sin " 4 cos = = 0 $ cos% + #$sin " = # sin " cos = 4 cos = ± =! 6 +!n, 5! 6 +!n CPM Educational Program 0 Chapter 6: Page 0 Pre-Calculus with Trigonometry
6-00.!!! + A!! + A!! A! A! = 0 A =! = + B B =! y 6-0. a. See graph at right. b. f () =, g() =, h() = + = + 6-0. 4 + cos z 4 + (! sin z) 4 +! sin z 7! sin z 6-0. a. log +9 ( ) + log 5 5 = 4 ( ) + = 4 ( ) = log +9 log +9 +9 = 9 = + 9 8 = 9 b. 500(.5)! + 000 = 0000 500(.5)! = 9000 (.5)! = 8! = log.5 8 = =.6807 = 0.840 log 8 log.5 = 9 8 CPM Educational Program 0 Chapter 6: Page Pre-Calculus with Trigonometry
Lesson 6.. 6-04. a. sin(! +!) b. sin(!) = sin(! +!) = sin! cos! + sin! cos! = sin! cos! 6-05. a. cos(! +!) b. cos(!) = cos(! +!) = cos! cos! " sin! sin! = cos! " sin! c. cos! = cos! " sin! = cos! " (" cos!) = cos! " + cos! = cos! " d. cos! = cos! " sin! = ( " sin!) " sin! = " sin! 6-06. a. sin cos = sin(! ) = sin 6 b. cos 40! + sin 40! = c. cos 40!! sin 40! = cos( " 40! ) = cos(80! ) d.! sin (y! 5) = cos((y! 5)) = cos(y! 0) e. sin 0! cos 40! + cos 0! sin 40! = sin(0! + 40! ) = sin(70! ) f. cos (w)! = cos( " w) = cos(4w) 6-07. a. sin cos = 4!sin cos = 4! sin cos = b. sin() = ( ) sin! (") = sin! " = # 6 + #n, 5# 6 + #n c. =! 6 +!n, 5! 6 +!n =! +!n, 5! +!n CPM Educational Program 0 Chapter 6: Page Pre-Calculus with Trigonometry
6-08. a. cos! = cos! + cos! = c. cos! cos! + ( ) = cos ( "! ) + cos(! ) = ± cos(!) + = cos(!) + b.! = " " =! d. cos! = " sin! sin! = " cos! sin! = sin! = ± " cos! " cos! sin( # ) = ± " cos# Review and Preview 6.. 6-09. sin = sin cos =! " 5! " 4 5 = 4 5 cos = sin " =!(" 4 5 ) " = 5 " = 7 5 sin = cos = " 6-0. sin = sin = "cos = "(" 4 5 ) = +cos = " +(" 5 ) = " ( ) sin! sin = sin! = 0.40!!!!! = "! 0.40 =.80 4 9 5 = 9 0 = 0 5 = " 0 6-. a. sin(! 5 p) = sin 0 p b.! sin " 4! " 6 ( ) =! sin ( "! " ) = sin (! " ) y 6-. See graph at right. (! )( + ) " 0 for!! " " CPM Educational Program 0 Chapter 6: Page Pre-Calculus with Trigonometry
6-. a. Any length such that 4.6 < AT < 0. The smallest!a = 0!"!AT = 0. The largest!a = 55!"!!T = 0!" AT = 4.6. b. AT = 4.6!or!AT! 0 c. AT < 4.6 6-4. sin! = 4 5!!!!!cos " = 5!!!!!sin " = # sec(! + ") = cos(! +") = cos! cos "#sin! sin " 6-5. = = 5 $(5 )#(4 5)(# ) 5 65+48 65 = 6 65 = 65 6 a. cos(.) + cos(0. +.) + cos(0.6 +.) + + cos(.7 +.) = cos(0.k +.) b. + 5 7 + 9 +!+ 0 = + (!) " + (!) " 5 + (!) " 7 +!+ 0 = # (!) n (n + ) 9! k=0 00 n=0 6-6. a.!! = 0 (! )( + ) = 0! = 0!!or!! + = 0 =!!or!! =! b. (! y ) + y + = 0! y + y + = 0 y! y! = 0!! "!!Same answer as part (a). Lesson 6.. 6-7. a. cos + sin + = 0 (! sin ) + sin + = 0! sin + sin + = 0! sin + sin + = 0 sin! sin! = 0 b. sin! sin! = 0 u = sin u! u! = 0 (u! )(u + ) = 0 u! = 0 or u + = 0 u = sin = or u = sin =! c. sin = is impossible since.5 is greater than. sin =!!!"!! = # CPM Educational Program 0 Chapter 6: Page 4 Pre-Calculus with Trigonometry
6-8. a. 8c! 4c = 0 4c(c! ) = 0 4c = 0 or c! = 0 c = 0 or c = 6-9. a. 8 cos = 4 cos 8 cos! 4 cos = 0 4 cos ( cos! ) = 0 cos = 0 or cos! = 0 = ", " or cos = b. s + s! = 0 (s + )(s! ) = 0 s + = 0 or s! = 0 s =! or s = b. sin + sin! = 0 (sin + )(sin! ) = 0 sin! = 0 or sin + = 0 sin = or sin =! = "!!!!!!!!!!!!!!!!!! 6-0. = ", 5" sin( +! ) + cos( +! ) = " cos sin cos! + cos sin! + cos cos! " sin sin! = " cos " cos + sin = " cos sin = 0 = n! 6-. a. The range of cos is! " ". b. You cannot divide cos by cos and you cannot cancel cos in the epression +cos. cos c. cos! cos! = 0 ( cos! )(cos + ) = 0 cos! = 0 or cos + = 0 cos = or cos =! Solutions: = " + "n 6-. a. sin = 0 or cos =! = 0, ", " b. =! " c. cos = 0 or tan =! = " 4, ", 5" 4, ", all + "n d. tan =! = ", 5" CPM Educational Program 0 Chapter 6: Page 5 Pre-Calculus with Trigonometry
Review and Preview 6.. 6-. y = +5! = (!)+9! = + 9! 6-4. a. cot (sec! cos ) = sin ( ) ( ) = cos sin " sin cos sin cos! cos ( )!cos ( cos ) =!cos sin = sin sin = Asymptotes at = and y =. b. cos! + sin = 0! sin + sin = 0 6-5. a. f () = b. g() = f ()! 6-6. y + = y! y = 6!!!!!6y " 6 = y! y = " 6" " = 6 6" (6 " ) = 6( " ) 6-7. See graph at right. a. y = +7!7 7 b.. lim +7!7 + -7. lim +7!" -7 " = 6 " 0 = 8 " 4 0 = 8( " ) =!!!y = # " = 6 = ". lim +7!7 " -7 = "# = 4. lim +7!"# -7 = CPM Educational Program 0 Chapter 6: Page 6 Pre-Calculus with Trigonometry
6-8. d = h!.6 h h = d(h!.6) d = h h!.6 6-9. a. sin! = 0 sin! cos! = 0 sin! = 0 cos! = 0! = 0, ", ", ", " b. sin! " cos! = 0 (sin! " cos!)(sin! + cos!) = 0 sin! " cos! = 0 sin! + cos! = 0 sin! = cos! sin! = " cos!! = # 4, # 4, 5# 4, 7# 4 6-0. sin =! 0, cos =! 7 7 sin = sin cos sin cos = "! 7 "! 0 0 = 7 49 cos = cos! ( )! = 40 49! 0 7 ( )! = 80 49! 49 49 = 49 6-. a. Eponential is reasonable if it really grows faster and faster. Linear fits well for this data but it does not fit her hypothesis. b. y = 5 c. y = 5 ( ), with = number of days since Monday. ( ) = 5 ( ) 4 =! 5065 ( ) = 9. y = 5 076 Perfect on Monday and Tuesday; 9. instead of 9 on Friday. It fits quite well. ( ) ( ) ( ) = ln ( 5 ) d. 00 = 5 5 = 5 ln 5 = 9.50 The following Wednesday night or Thursday early morning. y = 00 when = 9.50. CPM Educational Program 0 Chapter 6: Page 7 Pre-Calculus with Trigonometry
Lesson 6.4. 6-. b. Since cosine starts at a peak, we will not have to incorporate a horizontal shift. 6-. The period stays consistent regardless of the oscillations. 6-4. Half of the period. 6-8. No, the height of the oscillations will decrease with time. 6-9. 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. 6-40. The graph is approaching the vertical shift. Review and Preview 6.4. 6-4. Amplitude 5! = Vertical shift +.5 =.5 6-4. 6-4. Horizontal shift is right units Period 4 =! b!!"!!b =! y =.5 cos! ( " ) y =!.5 cos " ( ) +.5 or ( ) +.5 with a vertical flip instead of a horizontal shift cos =! 5, tan =, csc =! 5,!sec =!, cot = 5 5 sin! () :!# $! ", " % &, cos! () :![ 0, " ], tan! () :!(! ", " ) 6-44. tan! is inverse tangent while cot = tan. 6-45. 8m m = 6!!!!! 4m = 6!!!!!4m = 6!!!!!m = 9 CPM Educational Program 0 Chapter 6: Page 8 Pre-Calculus with Trigonometry
6-46. a.! + y = 00 y = 00 "! y = 00 "! b. A() = (00! " ) A() = 00! " 6-47. Draw a line through B parallel to CD meeting AC at E. Then AE = 60 cm, AB = 00 cm, and ABE is a right triangle. Hence BE = CD = 80. Let θ be the central BAC. Then cos θ = 0.6, so! " 0.97 radians. Thus the wire length around the large log is 80(π (0.97)) = 54.87 cm. The wire around the small log is 0((0.97)) = 7.09 cm in length and the wire between the logs is (80) cm. Thus, the total length is 54.87 + 7.09 + 60 = 55.79. C 0 E 60 θ 80 + 0 = 00 A B D 0 6-48. a. 5 (+)!= k (+) = k 5 +! = k 5 9 = k 5 k = 45! b. 6 (+k)!= 4 (+k) = 4 +k! = 4 k = k = Lesson 6.4. 6-49. a. y = k b. amplitude (a) c. The high points are decreasing while the low points are increasing. 6-50. The data looks surprisingly linear in the ZoomStat window. 6-5. a. slope = 6.55!6.746! =!0.9 y! 6.746 =!0.9(! ) y =!0.9(! ) + 6.746 y =!0.9 + 0.9 + 6.746 y = 6.99! 0.9 b. y =!0.9(9) + 6.99 y =!.77 + 6.99 = 5.0 y =!0.9(0) + 6.99 y =!.9 + 6.99 = 5.009 CPM Educational Program 0 Chapter 6: Page 9 Pre-Calculus with Trigonometry
6-5. a. It is half way between them. b. 6.746 =.75 + a! m am = 4.57 a = 4.57 m 4.57 m! 4.78 m = 0 4.57m! 4.78 = 0 4.57m = 4.78 m = 0.95778 c. y =.75 + (4.77)! 0.95778 p y =.75 + (4.77)! 0.95778 9 y =.75 +.7 = 5.4 a = 4.78 0.95778 = 4.77 6.55 =.75 + a! m 4.78 = am a = 4.78 m y =.75 + (4.77)! 0.95778 0 y =.75 +.0 = 5.75 6-5. Eponential decay is better. 6-54. a. The eponential function approaches the resting position of the spring. b..7 seconds c. p =.7 d. y = 4.77(0.95778).7! y = 4.77 (0.95778) " # y = 4.77(0.97494).7 $ % & e. y = 4.77(0.95778).7 cos! ( ) +.75.7 f..75. Students should say that the spring approaches the model s vertical shift. CPM Educational Program 0 Chapter 6: Page 0 Pre-Calculus with Trigonometry
Review and Preview 6.4. 6-55. sin a. (+cos )(+cos ) + = sin ++ cos +cos = sin (+cos ) sin (+cos ) sin (+cos ) b. (+cos ) = sin (+cos ) = cos!("sin! ) cos!(+sin! ) "sin! + sin = csc "sin! = cos! "cos! sin! +cos! +cos! sin! "sin! + cos sin (+cos ) = cos! cos! = cos! = sec! 6-56. a.! sin " = 0 sin " = sin " = ± " = # + #n, # + #n 6-57. a.!! 5 = 0 =!(!)± (!)!4()(!5) () = ± 44 ± ± = = 4 4 c. + = 7! ( + ) = 7! ( ) + = 49! 4 8 = 4 (8) = (4 ) 784 = 96 = 4 + b. 4 cos! = cos! = 4 cos! = ± 4 = ±! = " 6, 5" 6, 7" 6, " 6, all + "n b. 6 4!! 5 = 0 (6 + 5)(! ) = 0! = 0!!or!!6 + 5 = 0 =!or!!!!! "! 5 6 = ± d. + 5! = (!) (! ) + 5(! ) =! 4 + + 5! 5 = +! 5 = 0 (! 5)( + ) = 0 =!, 5 6-58. + ( + 4) = ( + 8) + + 8 + 6 = + 6 + 64! 8! 48 = 0 (! )( + 4) = 0 = (since "!4) The lengths of the sides of the triangle are, 6, and 0. CPM Educational Program 0 Chapter 6: Page Pre-Calculus with Trigonometry
6-59. " y =!7 cos ( (!.) ) + 5 6-60. 0.7 Amplitude:!8 = 4 =7 Period:.4 =! b!!"!!b =! 0.7 Horizontal shift:. (seconds) to the right Vertical shift: 5 (inches) up " =!7 cos ( (!.) 0.7 ) + 5 = 0.649,.5,.049,.55,.449,.95, 4.849 " 0.486 = cos ( (!.) ) 0.7.79 = " (!.) 0.7 0.5 =!. =.55 sin! cos! = cos! cos " + sin! sin " sin! cos! = # cos! sin! cos! + cos! = 0 cos!( sin! + ) = 0 sin! + = 0 or cos! = 0 sin! = #! = " + "n, " 6 + "n, 7" 6 + "n 6-6. a. y = k +6 = k +6 k = 7 y = f () = 7 +6 b. f (!) = 7!+6 = 7 f (0) = 7 0+6 = 7 6 ( ) = 7 ( ) = 7 f f a +6 = 7 9 = 7 " 9 = 9 a+6 = 7 a+6a a = 7 +6a a = 7 " a +6a = 7a +6a CPM Educational Program 0 Chapter 6: Page Pre-Calculus with Trigonometry
Closure Chapter 6 CL 6-6. a. b. 5π/6 π/6 0.5! /!! /! -0.5 c.! 6, the function sin! () can only have one output. - CL 6-6. a. cos =! cos =! = ", 4" b. sin = 4 sin = ± =!,! c. tan =! = " 4, 7" 4 CL 6-64. a. y = sin() +, y = cos! " 4 b. y =!sin! " 4 ( ( )) + ( ( ))!, y = cos ( + " 4 ) ( )! CL 6-65. a. See triangles at right. 6 b. sin 4! = 8 sin C 8 sin 4! = 6 sin C 4.475 6 = sin C 0.7456 = sin C A 8 cm 4 B 6 cm C B!C = 48. or!c =.79!!B = 80! " 4! " 48.! = 97.79!!B = 80! ".79! " 4! = 4.! A 8 cm 4 C 6 cm 6 sin 4! = AC sin 97.79! 6!sin 97.79! = AC!sin 4! 5.9446 = 0.559AC AC = 0.6 cm 6 sin 4! = AC sin4.! 6!sin4.! = AC!sin 4!.479 = 0.559AC AC =.6 cm c. If!B = 97.79º : If!B = 4.º : A = (6)(8) sin(97.79º ) =.78cm A = (6)(8) sin(4.º ) = 5.89cm Difference =.78 5.89 = 7.89 cm CPM Educational Program 0 Chapter 6: Page Pre-Calculus with Trigonometry
CL 6-66. ( ) =.7! b. tan! (!) =!6.4! a. tan! c. 80!! 6.4!!.7! = 8.9! CL 6-67. CL 6-68. sin! () : # $! ", " % &, cos! () :[ 0, " ], tan! () :(! ", " ) a. cos A = b. sin B = 5 c. cos(a + B) = cos A cos B! sin A sin B cos(a + B) = " 4 5 + 5 " 5 = 48 65! 5 65 = 65 CL 6-69. a. 0! sin cos = 0 sin(! ) = 0 sin(4) b. sin(! " ) = sin! cos " sin cos! c.! cos " + sin " =!(cos "! sin ") =! cos(") = 0 # cos " (") #sin = sin d. cos( +! ) = cos cos! " sin sin! = 0 #cos " ()#sin = " sin CL 6-70. a. (! sin ) + sin =! sin + sin = sin! sin = 0 sin ( sin! ) = 0 b. (! sin ) + sin =! sin + sin = sin! sin = 0 sin ( sin! ) = 0 sin = 0 or = 0, ", ", " 6, 5" 6 sin = 0 or = 0, ", ", " 6, 5" 6, all + "n Solution continues on net page. CPM Educational Program 0 Chapter 6: Page 4 Pre-Calculus with Trigonometry
CL 6-70. Solution continued from previous page. c. sin! sin() = 0 sin! sin cos = 0 sin (! cos ) = 0 sin = 0 cos = = 0, ", ", ", 5" d. sin! sin() = 0 sin! sin cos = 0 sin (! cos ) = 0 sin = 0 cos = = 0, ", ", ", 5", all + "n CL 6-7. Amplitude 6!.5 = 4.5 =.5 Period.5 =! b!!"!!b =!.5 = 0.8! Horizontal shift.5 seconds to the right Vertical shift.75 feet up h =.5 sin ( 0.8!(t ".5) ) +.75 a..055 feet b. At 0.704 and.5 0.704 =.96 seconds CL 6-7. a. 8(.0)! 0 = 00 8(.0) = 0 (.0) = 7.7778 log.0.0 = log.0 7.7778 = log 7.7778 log.0 = 97.64 c..7 = 60 8 = 0 (.7 ).7 = 0.7 =.0 CL 6-7. (!)! 4 = (!)a + b y =!! 6!5 =!a + b! 8 = a + b!7 = a + b b. log 5 log 5 ( ) = d. log +5! ( ) = 5 = 8 = 8 5 ( ) = ( ) = log +5!!7 = a + b!5 =!a + b! = b b =!6 +5! =! = + 5 = 8 = 4!7 = a! 6 a =! CPM Educational Program 0 Chapter 6: Page 5 Pre-Calculus with Trigonometry