Chapter 6: Extending Periodic Functions

Transcription

1 Chapter 6: Etending Periodic Functions Lesson 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! 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: = 0, + 4 =! 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 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

2 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! = 5! 6 6 5! 6 + 4! = 5! 6 + 4! = 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! a. sin! = sin = sin = b.! +!n,! +!n = ", " Review and Preview a. Since the string is 0 inches in length, the maimum point will be 0 inches above the minimum. b. 0 = 5 c = 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

3 6-. a. 0 = = = b. "! (k! ) + 7 left-sum = " #! (! ) + 7 $ % + " #! (! ) + 7 $ % + " #! (! ) + 7 $ % = 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 = 8 = = = 5.6 feet ! a. 0. = 6! 9! " = 6! " ! = 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 = ! 0. = 4.76! 9! " =.679! " 0. 9! = 4.56! = = liters/hr CPM Educational Program 0 Chapter 6: Page Pre-Calculus with Trigonometry

4 6-8. a.!!! =! (!) = (+)(!) (!) = + = = + = = b.!5 +!5!5 =!5!5 =! 5 =! 5 0 =!! 0 0 =! +!! 0 = (! ) ± =! = ± Lesson b. Inverses are symmetric about the line y =. c. No, because it does not pass the vertical line test 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

5 6-.! y! y!! " y = sin! () :!!D :[!,],!R : #! ", " % $ & y = cos! () :!!D :[!,],!R :[ 0, " ] 6-4. a b.! " 0.05 =.87 c !n,.87 +!n, for n an integer a. vertical line b..66 c =! ".66 d..66 +!n, !n or ±.66+!n, n an integer Review and Preview 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 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 6-9. See diagram at right. a.! "! b. 4! c. 4! d = = 5 = ± 5 cos! = " a. 6 = 0 + 8! (8)(0) cos = 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! = 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 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

7 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 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 a. 0 sin 8! = 0 sin I 0! = 0 sin I = sin I sin " = 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 sin 07.! d = d = 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 sin6.8! d = d =. m CPM Educational Program 0 Chapter 6: Page 7 Pre-Calculus with Trigonometry

8 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 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 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 sin 4! = 8 sin C 8! sin 4! = 9! sin C 4.47 = 9 sin C = sin C "C = sin "C = 9.8! ( ) # !B = 80! " 4! " 9.8! = 6.! 9 sin 4! = AC sin 6.! AC sin 4! = 9 #sin6.! AC = = 4.44 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 0 Chapter 6: Page 8 Pre-Calculus with Trigonometry

9 6-4. a. sin = 4 5 sin! 4 5 ( ) = 0.97 c pn,.4 + pn, n is an integer. b. = 0.97 and! " 0.97 = 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 " = y = + = + + y y+ = y++y y+ (y + ) = y + y + = y + y! y =! y(! ) =! y =!! =!! f! () =!! g() = (+)(! ) Asymptotes occur when the denominator equals zero. This occurs when = 0,!,. +cos! ("cos!)(+cos!) + "cos! ("cos!)(+cos!) +cos! +"cos! = "cos =! sin =! csc! f () = 7(9)! = ( )! =! = +! = + = () #%!(! ) +! for <! f (! )! = $ &% (! )!! for "! #%!(! ) +!!for <! h() = $ &% (! )!! for "! CPM Educational Program 0 Chapter 6: Page 9 Pre-Calculus with Trigonometry

10 Lesson 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,!,! 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 a. Restrict the range. b. Range: # $! ", " % & 6-5. a. lim!" tan # () = $ b. lim!"# tan" () = " $ 6-5. tan! = opposite adjacent = y tan! = ( ) tan " tan! = tan "! = 6.6! or 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

11 Review and Preview 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 Yes, the first is the inverse function, the second the reciprocal function of y = cos 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 =! = 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

12 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 =! 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 Laurel is. Hardy s equation only shifts the graph! 6 H () = sin! " ( ). ( ) = sin (! " 6 ) to the right since a. =! b. =! 6 c. H () = sin! " ( ) ( ) = sin (! " 6 ) y = sin((! " )) !! (!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

13 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. 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 y = sin! ( " ) (leg b) = 8 ( ) + (leg b) = 64! y 4 leg b = 9 a. sin! = 5 8 b. cos! = " 9 c. tan! = = 5 " # " 8 = " 5 # 9 = " 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, = 7. c. tan! tan = tan! ( 7 ) = or! " =.546 CPM Educational Program 0 Chapter 6: Page Pre-Calculus with Trigonometry

14 6-75. a.! 4! = 0 ( + )(! 7) = 0 =!, 7 c. + = 0 +! 0 = 0 (! 5)( + ) = 0 = 5,! b. (! )( + ) = 4!! = 4!! 6 = 0 (! )( + ) = 0 =!, d = ! 5 = 0 (! 5)( + 5) = 0 = 5,! 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 ( ) " a. y =! cos()! b. y = sin +! 4 c. y = sec() d. y = tan! h = kv r 5 = k = 0k k = h = V r h =!0 9 h = 0 9 = 0 CPM Educational Program 0 Chapter 6: Page 4 Pre-Calculus with Trigonometry

15 Lesson a. cos! " 4 b. cos! " 4 ( ) = cos sin ( " 4 ) + sin cos ( " 4 ) ( ) = cos # + sin # ( ) = (cos + sin ) ( ) = cos + sin c. cos! " 4 d. cos! " 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 = # = " 7 0 = "" 7 0 Review and Preview 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

16 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 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 = cos(! -") = cos! cos" + sin! sin" = #cos" + 0 $sin " = # cos" sin(! -") = sin! cos" # cos! sin" = 0 $ cos" + #(#) $ sin " = sin " 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 (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

17 Lesson y = 0 cos! ( " ) ! 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 " ) y = 0 cos(".09) + 44 y = y = 49.8 inches b. 7 tall = inches = 0 cos! ( " ) " 0 = cos! ( " ) 6 ( ) =! 6 cos " " = " ( " ) = 6.54 # 6 hours minutes pm " 6 hours minutes # 9 : 9 a.m 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 = h = cm b. = 4 cos!( ".5) + 4 cos " " 4 " = cos!( ".5) 4 ( ) =!( ".5).745! = " = ".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 # h = ft b. 7. = 4 cos! ( " ) = cos! ( " ) cos " (0.55) =! ( " ) = " =.47 sec = " =.58 sec CPM Educational Program 0 Chapter 6: Page 7 Pre-Calculus with Trigonometry

18 ( ) 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 = 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) = sin! (t ".5) sin " (0.5455) =! (t ".5) = 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) = cos ( 8! ( " 0.5) ) + 8 h = # cos " ( h = cm ) # 8! = " 0.5 = ( ) 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 = 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 = seconds.64 = t " 0, t = hours after noon or about :0 p.m.

19 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 " = sin ( 5! (t ".4) ) h =.5 cm sin " (" ) = 5! (t ".4) " = 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 = " = cm ( ) + ( (t " 5) ) b. 6 = 6 cos! (t " 5) 4 cos " 4 6 = cos! 4 ( ) =! 4 (t " 5).0709 = t " 5, t = seconds CPM Educational Program 0 Chapter 6: Page 9 Pre-Calculus with Trigonometry

20 Review and Preview a. Amplitude is 4. b. Horizontal shift is! 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 e = 5 " sin! = " 5 = " 5 tan! = " "4 = 4 csc! = 5 " = " 5 sec! = 5 "4 = " 5 4 cot! = "4 " = a.! b.!! = c.!! 5 =! " 5! = 0 d.!! 5 =! " 5! = 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

21 6-00.!!! + A!! + A!! A! A! = 0 A =! = + B B =! y 6-0. a. See graph at right. b. f () =, g() =, h() = + = 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 b. 500(.5)! = (.5)! = 9000 (.5)! = 8! = log.5 8 = =.6807 = log 8 log.5 = 9 8 CPM Educational Program 0 Chapter 6: Page Pre-Calculus with Trigonometry

22 Lesson a. sin(! +!) b. sin(!) = sin(! +!) = sin! cos! + sin! cos! = sin! cos! 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! 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) 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

23 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 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 = = 9 0 = 0 5 = " 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

24 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 < sin! = 4 5!!!!!cos " = 5!!!!!sin " = # sec(! + ") = cos(! +") = cos! cos "#sin! sin " 6-5. = = 5 $(5 )#(4 5)(# ) = 6 65 = 65 6 a. cos(.) + cos(0. +.) + cos(0.6 +.) + + cos(.7 +.) = cos(0.k +.) b !+ 0 = + (!) " + (!) " 5 + (!) " 7 +!+ 0 = # (!) n (n + ) 9! k=0 00 n= a.!! = 0 (! )( + ) = 0! = 0!!or!! + = 0 =!!or!! =! b. (! y ) + y + = 0! y + y + = 0 y! y! = 0!! "!!Same answer as part (a). Lesson 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

25 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

26 Review and Preview 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 = 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! 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

27 6-8. d = h!.6 h h = d(h!.6) d = h h! 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# sin =! 0, cos =! 7 7 sin = sin cos sin cos = "! 7 "! 0 0 = 7 49 cos = cos! ( )! = 40 49! 0 7 ( )! = 80 49! = 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 = 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 = CPM Educational Program 0 Chapter 6: Page 7 Pre-Calculus with Trigonometry

28 Lesson 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 Half of the period No, the height of the oscillations will decrease with time 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 The graph is approaching the vertical shift. Review and Preview Amplitude 5! = Vertical shift +.5 = 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! () :!(! ", " ) tan! is inverse tangent while cot = tan m m = 6!!!!! 4m = 6!!!!!4m = 6!!!!!m = 9 CPM Educational Program 0 Chapter 6: Page 8 Pre-Calculus with Trigonometry

29 6-46. a.! + y = 00 y = 00 "! y = 00 "! b. A() = (00! " ) A() = 00! " 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)) = 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 = C 0 E 60 θ = 00 A B D a. 5 (+)!= k (+) = k 5 +! = k 5 9 = k 5 k = 45! b. 6 (+k)!= 4 (+k) = 4 +k! = 4 k = k = Lesson a. y = k b. amplitude (a) c. The high points are decreasing while the low points are increasing The data looks surprisingly linear in the ZoomStat window a. slope = 6.55!6.746! =!0.9 y! =!0.9(! ) y =!0.9(! ) y =! y = 6.99! 0.9 b. y =!0.9(9) y =! = 5.0 y =!0.9(0) y =! = CPM Educational Program 0 Chapter 6: Page 9 Pre-Calculus with Trigonometry

30 6-5. a. It is half way between them. b =.75 + a! m am = 4.57 a = 4.57 m 4.57 m! 4.78 m = m! 4.78 = m = 4.78 m = c. y =.75 + (4.77)! p y =.75 + (4.77)! y = = 5.4 a = = =.75 + a! m 4.78 = am a = 4.78 m y =.75 + (4.77)! y = = Eponential decay is better a. The eponential function approaches the resting position of the spring. b..7 seconds c. p =.7 d. y = 4.77( ).7! y = 4.77 ( ) " # y = 4.77( ).7 $ % & e. y = 4.77( ).7 cos! ( ) 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

31 Review and Preview 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! a.! sin " = 0 sin " = sin " = ± " = # + #n, # + #n 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(! ) =! ! 5 = +! 5 = 0 (! 5)( + ) = 0 =!, ( + 4) = ( + 8) = ! 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

32 6-59. " y =!7 cos ( (!.) ) 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, " = cos ( (!.) ) = " (!.) =!. =.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 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

33 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 a. y = sin() +, y = cos! " 4 b. y =!sin! " 4 ( ( )) + ( ( ))!, y = cos ( + " 4 ) ( )! CL a. See triangles at right. 6 b. sin 4! = 8 sin C 8 sin 4! = 6 sin C = sin C = 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! = 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 = = 7.89 cm CPM Educational Program 0 Chapter 6: Page Pre-Calculus with Trigonometry

34 CL ( ) =.7! b. tan! (!) =!6.4! a. tan! c. 80!! 6.4!!.7! = 8.9! CL CL 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) = " " 5 = 48 65! 5 65 = 65 CL 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 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

35 CL 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 and =.96 seconds CL 6-7. a. 8(.0)! 0 = 00 8(.0) = 0 (.0) = log.0.0 = log = log log.0 = 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