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Chapter 6: Extending Periodic FunctionsLesson 6.1.16-1.a.The graphs of y sin x and y 12 intersect at many points, so there must be more than onesolution to the equation.b.There are two solutions. From the graph we can see y !6 and y 56! .c.It shows where the y-coordinate or sin x 0.5 .x 43! and x 53! . Students may use unit circle or the graph.d.6-2.Draw a vertical line at x 12 . The angles that satisfy the equation are x 6-3.6-4.and x 5!3.A horizontal line drawn at y 2 does not intersect the unit circle. The value 2 is not in therange for y sin x .Examples of trig equations: cos x 3 , csc x 0Examples of non-trig equations: x 2 3x 4 0 , 3x 4 2x ! 1 x6-5.sin x 1 0a.b.sin x !1x c.!3cos x !3"2x cos x ! 2x 3"42 cos x !1d., 54"122" 4 ", 332 sin x ! 3 02 sin x 3sin x x 32" 2",3 36-6.a.All real numbers.b.!1 " y " 1c.The functions both have a period of 2! , so a shift of that size would not affect eitherfunction.6-7.a.b.c.d.There would be an infinite number of solutions.2 solutions: 0 and πInfinitely many.An integer multiple of 2! , because it is the period ( 2! n for n an integer).CPM Educational Program 2012Chapter 6: Page 1Pre-Calculus with Trigonometry

6-8.a.There are an infinite number of solutions. There are 12 solutions on the graph given.5!b.c.Add 2! n to !6 , 5!, n is any integer. 2! 56! 126! 176!66d.5!" 2! 56! " 126! " 76!6! 4! !6 246! 256!65! 4! 56! 246! 296!6e.! "6 ! 5" ! "6 !!5"6! 5" !30" 65"! 306"6! 31"6! 356"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!or! 56! .66-10.2 sin x ! 3 a.b.!3 2! n,2!3 2! n2 sin x 3sin x x 32" 2",3 3Review and Preview 6.1.16-11.a.Since the string is 30 inches in length, the maximum point will be 30 inches above theminimum.30 1515 5 20b.c.2d.2!2.5f.h !15 cos 2!52 2! " 25 4!5e.! cos x( 45" (t) ) 206-12.a.csc 56! b.tanc.cotd.sec1 112 1" 21 2sin 5! 6sin ! 2! cos ! /2 10 " undefined2cos 5! 3125! sin 5! 3 12 # 23" 3 2" 37!6 1sin 7! 6 1" 3 2CPM Educational Program 2012 1#2" 3 1" 32" 3 " 33"2 33Chapter 6: Page 2Pre-Calculus with Trigonometry

6-13.a.3x0 1x1 2x2 3b." ! 12 (k ! 1)2 7k 1left-sum "# ! 12 (1 ! 1)2 7 % "# ! 12 (2 ! 1)2 7 % "# ! 12 (3 ! 1)2 7 % 7 13 526-14.3b ! a55b 3b ! a a 2bx ! 555x b!y3(b ! y) ! 2y a3b ! 3y ! 2y a!5y a ! 3b3b ! ay 5x b!6-15.45 x32128 5x1285 xx 25.6 feet6-16.a.9!0.2 36!x9! " x 36! " 0.29! x 7.2!x 7.29V 13 ! " 6 2 "10 120!b.12"V 12 "120! 60!60! 13 ! r 2 " h 0.8 liters/hr180 r 2( 53 r )108 r 3r 4.76226-17.g(!1) (!1)2 ! 2(!1)g(!1) 1 2 3g(3) 32 ! 2(3)g(3) 9 ! 6 3g(a) a 2 ! 2(a)9!0.2 4.7622 2 !x9! " x 22.679! " 0.29! x 4.536!x 4.5369 0.504 liters/hrg(a) a 2 ! 2ag(t ! 2) (t ! 2)2 ! 2(t ! 2)g(t ! 2) t 2 ! 4t 4 ! 2t 4 t 2 ! 6t 8CPM Educational Program 2012Chapter 6: Page 3Pre-Calculus with Trigonometry

6-18.a.x22x !x!1x2 ! xx 2 !1x(x!1) 3(x 1)(x!1)x(x!1) 3x 1x 3b. 3 x!5x 2 !252 x!5x 2 !25 1 12x ! 5 x 2 ! 250 x 2 ! 2x ! 200 x 2 ! 2x 1 ! 1 ! 203x x 12x 1x xx 2 !2521 (x ! 1)2 21 x ! 112x 1 21Lesson 6.1.26-19.b.Inverses are symmetric about the line y x .c.No, because it does not pass the vertical line test.6-20.b.# ! "2 , "2 % &The domain of y sin !1 x will be the range of y sin x , so the domain is [ !1, 1] .6-21.a.!3a.b.c.d. 1.047It is not in the range of y sin !1 x . The inverse of sine only selects one of the infinitelymany solutions to the equation.x !3 2! n or 23! 2! nYou 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 ofy cos!1 x is [ 0, ! ] .CPM Educational Program 2012Chapter 6: Page 4Pre-Calculus with Trigonometry

!6-23.!2!2x–1yy1x!"2–1y sin !1 (x) :!!D : [ !1, 1] ,! R : # ! "2 , "2 %&1y cos!1 (x) :!!D : [ !1, 1] ,! R : [ 0, " ]6-24.0.305a.b.! " 0.305 2.837c.0.305 2! n, 2.837 2! n , for n an integer.6-25.a.b.c.d.vertical line1.2665.017 2! " 1.2661.266 2! n, 5.0177 2! n or 1.266 2!n , n an integerReview and Preview 6.1.26-26.a.It is not in the range of y cos!1 x . cos!1 x selects only one of the infinitely manysolutions to the equation.x !3 2! n or 53! 2! nb.c.You have to draw and think.6-27.tan x sin xcos x 0 ! sin x 0x " , 2" , 3" , 4" x n" , n is any integer6-28.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.CPM Educational Program 2012Chapter 6: Page 5Pre-Calculus with Trigonometry

6-29.a.b.c.d.See diagram at right.! "3!43!4!66-30.2 2 x 2 32x2 5x 5cos ! "6-31.a.536 2 10 2 8 2 ! 2(8)(10) cos x36 164 ! 160 cos x!128 !160 cos x0.8 cos xb.xsin 60!x sin 70! 28sin 70! 28 sin 60!0.9397x 24.2587x 25.8cos!1 0.8 cos!1 (cos x)x 36.9!6-32.a.log 2( 641 ) log2 ( 64 !1 )!1 log 2 ( 2 6 ) log 2 2 !6 !6b.log 8 1 0c.log 8 81 1d.log 2 (64) log 2 (2 6 ) 6e.impossiblef.log 5 251 3 log 5b.(x h)2 ! x 2h()(( 52 )1 3 ) log 5 ( 5 2 3 ) 236-33.a.2 x 3 y2 ! 4 x 2 y2 2 xy23xy 3 ! 3y 3 2 xy2 (x 2 !2 x 1)3y 3 (x!1) 2 x(x!1)(x!1)3y(x!1) 2 x(x!1)3yCPM Educational Program 2012 Chapter 6: Page 6x 2 2 xh h 2 ! x 2h22 xh hhh(2 x h) 2x hhPre-Calculus with Trigonometry

6-34.a.! f (x) 2Flipped over x-axis and up 2.b.2 f (!x)Flipped over y-axis and stretchedvertically.yyxxc.1f (x)Asymptotes at x !2, 0 , and 2.yxLesson 6.3.16-35.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 Ib.20sin 28!30 ! 0.4695 20 sin I14.08520 sin Ic.dsin 107.2d 19.20560.469544.8! #I(or !I d # 0.4695 20 sin 107.2!sin "1 0.70425 sin "1 sin I 135.2! ,!D 180! " 28! " 44.8! 107.2!d 40.69 mbut don ot point this out yet)Katya missed the possibility that !I could be obtuse. !I 180! " 44.8! 135.2!!D 180! " 135.2! " 28! 16.8!20sin 28! dsin 16.8!d # 0.4695 20 sin 16.8!d 5.78060.4695d 12.31 mCPM Educational Program 2012Chapter 6: Page 7Pre-Calculus with Trigonometry

6-37.a.See diagram at right. The horizontal line crosses the unitcircle at two different angles.b.Inverse sine has a restricted range, which does notinclude the 2nd quadrant.6-38.a.10sin 90!1 !102 asin 30!b.10sin C 3sin 30!30! 10 ! sin5 3 sin C53d.3 ! sin C" sin C5 ! sin C"C 90!10 7sin C10 ! sin57is [ #1, 1] .6-39.5sin 30!30! sin 30!30! 7 ! sin C5 7 sin CNot possible since the range of sinee.f.g. 10 ! sin5 5 sin C1 sin C a !1a 5c.10sin C sin C"C sin #1( 57 )"C 45.58!or "C 180! # 45.6! 134.4 !!ACB 180! " !BCC # 180! " !BC #C since !BCC " is isosceles.Supplementary angles have the same sine.One triangle.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 6.3.16-40.9sin 34! !B 180! " 34 ! " 29.8! 116.2!8sin C9sin 34!8 ! sin 34 ! 9 ! sin C4.47 9 sin C4.479AC sin sin C"C sin #1 AC ( 4.479 )ACsin 116.2!34 ! 9 # sin 116.2!8.07530.5592 14.44 cm"C 29.8!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 2012Chapter 6: Page 8Pre-Calculus with Trigonometry

6-41.sin x a.sin !1c.45b.( 45 ) 0.927x 0.927 and ! " 0.927 2.2140.9273 2 pn, 2.2143 2 pn , n is an integer.6-42.g(x) g(6) 19.22g(!3) 19.2(!3)21.2 g(6) g(!3) 96 " 15 9kx2k4269651 8!1 8! 365 315 32 " 15 3 3215k 16 !1.2 19.26-43.y 1 x 1 x x(y 2) 2y 2xx 2yy 2xy 2x 2y 2xy ! 2y 2 ! 2xy(x ! 2) 2 ! 2xy 2 yy 2y f !1 (x) 2!2 xx!22 x!22! x 2 x!22! x6-44.1g(x) 1x(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"cos1"cos 1 cos ! 1"cos ! )(1 cos ! )21"cos !6-46.1 x!1f (x) 27(9) 2(2 1 x!12 3332sin 2 ! 2 csc 2 !) 333x!2 33 x!2 3x 1 3(3)x6-47.#% !2(x ! 3) 3 ! 2f (x ! 3) ! 2 %& 2(x ! 3) ! 1 ! 2#% !2(x ! 3) 1h(x) %& 2(x ! 3) ! 1 ! 2CPM Educational Program 2012for x 1 ! 3for x " 1 ! 3!!for x !2for x " !2Chapter 6: Page 9Pre-Calculus with Trigonometry

Lesson 6.1.46-48.a.You would find vertical asymptotes when cos x 0 . These occur at x ! 32" , ! "2 , "2 , 32" .b.This would be when the graph of tan x crosses the x-axis, which are the roots, and theyoccur at x 2! , "! , 0, ! , 2! .6-49.x!a.c.n"2, where n is any odd integer.b.All real numbers.y 0, x ! n , n is any integer.d.x Restrict the range.b.Range: # ! "2 , "2 %&n!2, where n is any odd integer.6-50.a.6-51.a.lim tan #1 (x) x!" 2b.lim tan "1 (x) " 2x!"#6-52.tan ! oppositeadjacenttan ! 12 yx6-53.tan "1 tan ! tan "1( 12 )! 26.6! or 0.464 radians6-54.452adjacent side tan ! oppositeadjacent 22.5822.5tan "1 tan ! tan "1( 22.58 )! 19.573!6-55.! 1.2 radianstan 1.2 2.572approximate slope 2.572CPM Educational Program 2012Chapter 6: Page 10Pre-Calculus with Trigonometry

Review and Preview 6.1.46-56.2 sin x ! 1 0a.2 sin x 1sin x x c.12"6b.cos x d.!2 cos x ! 2x 6-57.6-58.22" 2" n, 34"4!22 !1x " 2" n, n is an integer 2" n, 56" 2" n, n is an integer2 ! 2 sin x 0cos x 2 2 cos x 02 cos x !2cos x 3.8 0cos x !3.8cos x 1/ " no solution 2" n, n is an integerYes, the first is the inverse function, the second the reciprocal function of y cos x .sin x 0.3 has infinite solutions unless we are working with a restricted values of x. Theexpression 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( !3 !6 ) sin ( 26! !6 ) sin ( !2 ) 1but sin ( !3 ) sin ( !6 ) 23 12 3 1"12. sin6-60.2x 2 ! 8x a 2(x 2 2xb b 2 )!2x 2 ! 8x a 2x 2 4xb 2b 28 4bb 2a !2b 2 2 ! 2 2 86-61.Amp. 3, horizontal shift 2 to the right, vertical shift 1 up, period 2!! 2 2!1" !2 4 .6-62.tan 23! sin 2! 3cos 2! 3 3 2"1 2CPM Educational Program 2012 32#"21 " 3Chapter 6: Page 11Pre-Calculus with Trigonometry

6-63.a.2!6 slope of PR 14!(!4)perpendicular slope midpoint of PR y!4 y perpendicular slope y ! 12 y 6-64. !29b.4 slope of median 12!2!58!3 !83y ! 4 ! 83 (x ! 5)92, 6 2 (5, 4)( !4 1022 )y ! 83 (x ! 5) 4(x ! 5)9292(x ! 5) 42!6 slope of PR 14!(!4)c.!418929292!418 !29(x ! 2)(x ! 2) 12x0 1.25, x1 1.75, x2 2.25, x3 2.75,!x4 3.25, x5 3.75, x6 4.25, x7 4.75xk 0.5k 1.25sum 127! 0.5k 1 1.25 " 1.600k 0Lesson 6.2.16-65.Laurel is. Hardy’s equation only shifts the graph((()H (x) sin 3x ! "2 sin 3 x ! "6)) .!66-66.a.6-67.6-68.a.b.x !2b.x !6c.to the right since()((H (x) sin 3x ! "2 sin 3 x ! "6))y 2 sin(3(x ! " )) 4!1 ! (!5) 22Horizontal shift is !2 to the right. Vertical shift is 3 down. The period isAmplitude ((y 2 sin 2 x ! "2)) ! 3CPM Educational Program 2012Chapter 6: Page 122!2 ! .Pre-Calculus with Trigonometry

6-69.a.y 3 cos(! (x 1)) " 2y 2 sinb.( ( x ! ))13"26-71.a.(0.4, 46) and (2.2, 26)b.Vertical shift 26 c.26 ! 46 20 10 , horizontal shift 0.4 or –1.4,22Period 2(2.2 ! 0.4) 3.6 , Amplitude 202 26 10 36 .One possible answer is h(t) 10 cos( ( ) (t " 0.4) ) 36.2!3.6yReview and Preview 6.2.16-72.y 3 sin(!242)(x " 2) 1x6-73.–25 2 (leg b)2 8 224–4(leg b)2 64 ! 25leg b 39a.sin ! b.cos ! "c.tan ! 5839858" 39 8 58 # "839 "539#3939 "5 39396-74.a.The range of sine and cosine is !1 " y " 1 .b.A fraction can equal 73 without the numerator being 3 and the denominator being 7. For0.3 3example, 0.7.73!1!1tan tan x tanc.7( )x 0.405 or! 0.405 " 3.546CPM Educational Program 2012Chapter 6: Page 13Pre-Calculus with Trigonometry

6-75.a.x 2 ! 4x ! 21 0b.(x 3)(x ! 7) 0x !3, 7x2 ! x ! 2 4x2 ! x ! 6 0(x ! 3)(x 2) 0x !2, 33x 2 x 10c.(x ! 2)(x 1) 46x 2 5x 25d.3x 2 x ! 10 0(3x ! 5)(x 2) 06x 2 5x ! 25 0(3x ! 5)(2x 5) 0x 53 , !2x 53 , !526-76.tan x!csc xsec x sin x ! 1cos x sin x1cos x 1cos x1cos x 16-77.tan 28! 0.532y ! 912 0.532(x ! 285)6-78.sec x!tan xsin x 1 ! sin xcos x cos xsin x 6-79.y !3 cos(2x) ! 1a.c.y sec(x)sin xcos2 xsin x sin x ! 1cos2 x sin x sec 2 x1cos2 x()b.y 2 sin x !4 " 2d.y tan !1 x6-80.h 15 kVr2h 3Vr2k204h 3!10930 109360 20kk 3CPM Educational Program 2012h Chapter 6: Page 14Pre-Calculus with Trigonometry