Cumulative Review 161

4. Given that f(x = X2 + 3x and g(x) = 5x + 3, solve f(x) = g(x). Graph each function and label the points of intersection.

5. Graph f(x) = ,- 3)2 - 2 using transformations.

6. For the quadrat : function f(x) = 3x2 - 12x + 4,(a) Determine .hether the graph opens up or down.(b) Determine ie vertex.(c) Determine ie axis of symmetry.(d) Determine ie intercepts.(e) Use the inf rmation from parts (a)-(d) to graphf

7. Determine whei er f(x) = _2x2 + 12x + 3 has a maximum or minimum value. Then find the maximum or minimum value.

8. Solve x2 - lOx - 24 ~ O.

9. HV Rental n weekly rental cost of a 20-foot recreational vehicle is $129.50 plus $0.15 per mile.(a) Find a line; function that expresses the cost C as a function of miles driven m.(b) What is the ental cost if 860 miles are driven?(c) How many riles were driven if the rental cost is $213.80?

10. The variable int rest rate on a student loan changes each July 1 based on the bank prime loan rate. For the years 1992 to 2004, thisrate can be app rximated by the model

rex) = -0.115x2 + U83x + 5.623

where x is the n rnber of years since 1992 and r is the interest rate as a percent.Source: u.s. Fee rat Reserve(a) During whi 1 year was the interest rate the highest? Determine the highest rate during this time period.(b) Use the me el to estimate the rate in 2010. Does this value seem reasonable?

CUMULATIV REVIEW

10. Is the following graph the graph of a function?1. Find the distal 'e between the points P = (-1, 3) andQ = (4, - 2). Fi 1 the midpoint of the line segment P to Q.

2. Which of theollowing points are on the graph ofy = x3 - 3x + ?(a) (-2,-1) (b) (2,3) (c) (3,1)

3. Solve the inequ: ity 5x + 3 ~ 0 and graph the solution set.

4. Find the equati: l of the line containing the points (-1, 4)and (2, -2). Ex ress your answer in slope-intercept formand graph the Ii "

5. Find the equat -n of the line perpendicular to the liney = 2x + 1 ane :ontaining the point (3, 5). Express youranswer in slope- rtercept form and graph the line.

6. Graph the equa rn x2 + i -4x + 8y - 5 = O.

7. Does the foil wing relation represent a function?{( -3,8), (1,3), Z. 5), (3,8)}

defined by f(x) = x2 - 4x + 1, find:8. For the function(a) f(2)(b) f(x) + f(2:(c) fe-x)(d) -f(x)(e) f(x + 2)

f(x + h) -(f) h

9. Find the domain of h(z) = 3z - 1.6z - 7

(x)h*O

x

11. Consider the function f(x) = _x_.x+4

(a) Is the point ( 1, ~) on the graph of f?

(b) If x = -2, what is f(x)? What point is on the graphoff?

(c) If f(x) = 2, what is x? What point is on the graphoff?

X212. Is the function f( x) = --- even, odd, or neither?

2x + 1

13. Approximate the local maxima and local minima off(x) = x3 - 5x + Ion (-4, 4). Determine where thefunc-tion is increasing and where it is decreasing.

14. If f(x) = 3x + 5 and g(x) = 2x + 1,(a) Solve f(x) = g(x).(b) Solve f(x) > g(x).

U-k lice ~l Teo. ~ d4l l<estdlLD ~ MUa...r- SECTION 4.5 The Real Zeros of a Polynomial Function 227e1.. r-es~ ( I..U).y\ci() {:'C~) ~ dLu-lc:tc. fV1 t')(-c...-

13. f(x) = 3x4 - x3 - 5x + 10; x - 2 14. f(x) = 4X4 - 15x2 - 4; x - 2

15. f(x) = 3x6 + 2x3 + 27; x + 3 16. f(x) = 2x6 - 18x4 + x2 - 9; x + 3

17. f(x) = 4x6 - 4X4 + X2 - 15; x + 4 18. f(x) = x6 - 16x4 + x2 - 16; x + 4

19. f(x) = 2x4 - 3 + 2x - 1; x - ~1

20. f(x) = 3x4 + x3- 3x + 1; x + "3

In Problems 21-32, ill the maximum number of real zeros that each polynomial function may have. Then use Descartes' Rule of Signsto determine how n ny positive and how many negative zeros each polynomial function may have. Do not attempt to find the zeros.

'21. f(x) = -4x7. x3 - x2 + 2 22. f(x) = 5x4 + 2x2 - 6x - 5 23. f(x) = 2x6 - 3x2 - X + 1

24. f(x) = -3xs. 4X4 + 2 25. f (x) = 3x3 - 2x2 + X + 2 26. f (x) = - x3 - X2 + X + 1

27. f(x) = -x4 + ~2 - 1 28. f(x) = X4 + 5x3 - 2 29. f(x) = xS + x4 + x2 + X + 1

30. f (x) = xS - J + x3 - x2 + x-I 31. f(x) = x6 - 1 32. f (x) = x6 + 1

In Problems 33-44, 'st the potential rational zeros of each polynomial function. Do not attempt to find the zeros.

-,~. f(x) = 3x4 - x3 + X2 - X + 1 34. f(x) = xS - x4 + 2x2 + 3 35. f(x) = xS - 6x2 + 9x - 3

~. f(x) = 2xs - 4 - x2 + 1 37. f(x) = -4x3 - X2 + X + 2 38. f(x) = 6x4 - x2 + 2

39. f(x) = 6x4 - 2 + 9 40. f(x) = -4x3 + x2 + X + 6 41. f(x) = 2xs - x3 + 2x2 + 12

42. f(x) = 3xs - 2 + 2x + 18 43. f(x) = 6x4 + 2x3 - X2 + 20 44. f(x) = -6x3 - x2 + X + 10

In Problems 45-56, se the Rational Zeros Theorem to find all the real zeros of each polynomial function. Use the zeros to factor f overthe real numbers.

'45. f(x) = x3 + 2 2 - 5x - 6 46. f(x) = x3 + 8x2 + llx - 20 47. f(x) = 2x3 - x2 + 2x - 1

48. f(x) = 2x3 + 2 + 2x + 1 49. f(x) = 2x3 - 4x2 - lOx + 20 50. f(x) = 3x3 + 6x2 - 15x - 30

51. f(x) = 2X4 + 3 - 7x2 - 3x + 3 52. f(x) = 2X4 - x3 - 5x2 + 2x + 2 53. f(x) = X4 + x3 - 3x2 - X + 2

54. f(x) = x4 - J - 6x2 + 4x + 8 55. f(x) = 4X4 + 5x3 + 9x2 + lOx + 2 56. f(x) = 3x4 + 4x3 + 7x2 + 8x + 2

In Problems 57-68, olve each equation in the real number system.

'57. x4 - x3 + 2x2 - 4x - 8 = 0 58. 2x3 + 3x2 + 2x + 3 = 0

59. 3x3 + 4x2 - 7. + 2 = 0 60. 2x3 - 3x2 - 3x - 5 = 0

61. 3x3 - x2 - 15 + 5 = 0 62. 2x3 - llx2 + lOx + 8 = 0

63. x4 + 4x3 + 2x - x + 6 = 0 64. x4 - 2x3 + lOx2 - 18x + 9 = 0

3 2 2 865. x -"3 x +"3. + 1 = 0

366 x3 + - x2 + 3x - 2 = 0• 2

67. 2x4 - 19x3 + : 'x2 - 64x + 20 = 0 68. 2x4 + x3 - 24x2 + 20x + 16 = 0

228 CHAPTER 4 Polynomial and Rational Functions

In Problems 69-80, graph each polynomial function.

69. f(x) = x3 + 2X2 - 5x - 6 70. f(x) = x3 + 8x2 + llx - 20

72. f(x) = 2x3 + x2 + 2x + 1 73. f(x) = x4 + x2 - 2

75. f(x) = 4X4 + 7x2 - 2 76. f(x) = 4x4 + 15x2 - 4

78. f(x) = X4 - x3 - 6x2 + 4x + 8 79. f(x) = 4x5 - 8x4 - X + 2

71. f(x) = 2x3 - X2 + 2x -

74. f(x) = X4 - 3x2 - 4

77. f(x) = X4 + x3 - 3x2 - + 2

80. f(x) = 4x5 + 12x4 - X - 3

In Problems 81-88, find bounds on the real zeros of each polynomial function.'81. f (x) = X4 - 3x2 - 4 82. f (x) = X4 - 5x2 - 36

83. f(x) = x4 + x3 - x-I 84. f(x) = x4 - x3 + x-I

85. f(x) = 3x4 + 3x3 - x2 - 12x - 12 86. f(x) = 3x4 - 3x3 - 5x2 + 27x - 36

87. f(x) = 4x5 - x4 + 2x3 - 2x2 + x-I 88. f(x) = 4x5 + X4 + x3 + x2 - 2x - 2

In Problems 89-94, use the Intermediate Value Theorem to show that each polynomial function has a zero in the 'ven interval.'89. f(x) = 8x4 - 2x2 + 5x - 1; [O,lJ 90. f(x) = X4 + 8x3 - x2 + 2; [-1, OJ

91. f(x) = 2x3 + 6x2 - 8x + 2; [-5, -4J 92. f(x) = 3x3 - lOx + 9; [-3, -2J

93. f(x) = x5 - x4 + 7x3 - 7x2 - 18x + 18; [1.4,1.5J 94. f(x) = x5 - 3x4 - 2x3 + 6x2 + X + : [1.7,1.8J

In Problems 95-98, each equation has a solution r in the interval indicated. Use the method of Example 10 to app. -ximate this solutioncorrect to two decimal places.

95. 8x4 - 2x2 + 5x - 1 = 0; O:=; r :=; 1

97. 2x3 + 6x2 - 8x + 2 = 0; -5:=; r :=; -4

96. x4 + 8x3 - x2 + 2 = 0; -I:=; r :=;0

98. 3x3 - lOx + 9 = 0; - 3 :=; r :=; - 2

In Problems 99-702, each polynomial function has exactly one positive zero. Use the method of Example 10 to iproximate the zerocorrect to two decimal places.

99. f(x) = x3 + x2 + X - 4

101. f(x) = 2X4 - 3x3 - 4x2 - 8

Applications and Extensions

100. f(x) = 2x4 + x2 - 1

102. f(x) = 3x3 - 2x2 - 20

103. Find k such that f(x) = x3 - kx2 + kx + 2 has thefactor x - 2.

104. Find k such that f(x) = x4 - kx3 + kx2 + 1 has thefactor x + 2.

105. What is the remainder when f(x) = 2x20 - 8xlO + X - 2is divided by x-I?

106. What is the remainder when f(x) = -3X17 + x9 - x5 + 2xis divided by x + I?

107. Use the Factor Theorem to prove that x - c is a factor ofX" - cn for any positive integer n.

108. Use the Factor Theorem to prove that x + c is a factor ofxn + c" if n 2: 1 is an odd integer.

109. One solution of the equation x3 - 8x2 + 16x - 3 = 0 is 3.Find the sum of the remaining solutions.

110. One solution of the equation x3 + 5x2 + 5x - 2 = 0 is -2.Find the sum of the remaining solutions.

111. Geometry What is the length of the edge of a cube if, aftera slice 1 inch thick is cut from one side, the volume remain-ing is 294 cubic inches?

112. Geometry What is the length of the edge of a cube if itsvolume could be doubled by an increase of 6 centimeters inone edge, an increase of 12 centimeters in a second edge, anda decrease of 4 centimeters in the third edge?

113. Let f(x) be a polynomial function whetegers. Suppose that r is a real zero of]coefficient of f is 1. Use the Rational Z(that r is either an integer or an irration

114. Prove the Rational Zeros Theorem.

[Hint: Let E., where p and q have no ccq

1and -1, be a zero of the polynomial

f(x) = anx" + al1_,x"-' + ..

whose coefficients are all integers. Sho

anP" + a,,_,p"-'q + ... + a,pq'

Now, because p is a factor of the firsttion,p must also be a factor of the terrrfactor of q (why?),p must be a factor 0

be a factor of awl115. Bisection Method for Approximating,

We begin with two consecutive integethat f(a) and f(a + 1) are of oppositthe midpoint ml of a and a + 1. If f(mzero of f, and we are finished. Otherwsite sign to either f(a) or f(a + 1). Sland f(mj) that are of opposite sign. l\midpoint m2 of a and m,. Repeat tl

~coefficients are in-and that the leadingos Theorem to showInumber.

irnon factors except

nction

+- alx + ao

that, + aoqn = 0

terms of this equa-loqn. Since p is not aao. Similarly, q must

-ros of a Function f5, a and a + 1, suchsign. Evaluate f atI = 0, then ml is thee,f(mj) is of oppo-ipose tha tit is f (a).w evaluate f at thes process until the

In Problems 39-44, 1

39. 10g< U~2), U'ite each expression as the sum and/or difference of logarithms. Express powers as factors.

( . r.)4 41. log(x2~),. 0, V > 0, w > 0 40. IOg2 a2 vb, a > 0, b > 0 x>O

Chapter Review 337

In Problems 15 and 6, state why the graph of the function is one-to-one. Then draw the graph of the inverse function r'. For conve-nience (and as a hin . the graph of y = x is also given.

15. y

4 ;t,=x

(3,3)

(2,0) 4 x

16. y

4 y=x

(2,1)

-4 (1,0) 4 x

\q,-1)

-4

-4

In Problems 17-22,range of f and Fl.

2x + 317. f(x) = -5 -x - 2

ie function f is one-to-one. Find the inverse of each function and check your answer. Find the domain and the

2 - x18. f(x) =--

3 + xI

19. f(x) =--x-I

20. f(x) = ~3

21. f(x) = 1/3x

22. f(x) = XI/3 + 1

In Problems 23 and 4, f(x) = 3x and g(x) = log, x.

23. Evaluate: (a) (4) (b) g(9) (c) f( -2) (d) g(:7)

24. Evaluate: (a) ( 1 ) (b) g(81) (c)f(-4) (d) g(2!3)

In Problems 25 andconvert each logaritl

25. 52 = Z

5, convert each exponential expression to an equivalent expression involving a logarithm. In Problems 27 and 28,tic expression to an equivalent expression involving an exponent.

26. as = m 27. IOg5u = 13 28. loga4 = 3

In Problems 29-32,., id the domain of each logarithmic function.

29. f(x) = log(3x 2) 30. F(x) = logs(2x + 1) 31. H(x) = log2(x2 - 3x + 2) 32. F(x) = In(x2 - 9)

In Problems 33-38" aluate each expression. Do not use a calculator.

33. IOg2( i) 34. IOg381 35.lneV2

y'338. IOg22

(X2 + 2x -l 1)

42. logs x2- , X > 0 (2x + 3 )2

44. In 2 ' x> 2x - 3x + 2

In Problems 45-50, 1 'ite each expression as a single logarithm.

145. 3 log, x2 + 2 10 . \IX

47. Ine : 1) + h :x: I) - In(x2 - 1)

1 .49. 21og2 + 3log. - 2[log(x + 3) + log(x - 2)]

48. log(x2 - 9) - log(x2 + 7x + 12)

1 1 1SO. 21n(x2 + 1) - 41n2 - 2[ln(x - 4) + lox]

In Problems 51 and: . use the Change-of-Base Formula and a calculator to evaluate each logarithm. Round your answer to three dec-imal places.

51. IOg419 52. log, 21

338 CHAPTER 5 Exponential and Logarithmic Functions

~11n Problems 53 and 54, graph each function using a graphing utility and the Change-of-Base Formula.53. y = log, x 54. y = log, x

In Problems 55-62, use the given function f to:(a) Find the domain of!(b) Graph!(c) From the graph, determine the range and any asymptotes of!(d) Findr\ the inverse of!(e) User I to find the range of!(f) Graphrl.

55. f(x) = 2x-3 56. f(x) = -2x + 3I

57 f(x) = -(TX) 58. f(x 1 + rx. 2

59. f(x) = 1 - e-x 60. f(x) = 3ex-21

61. f(x) = Zln(x + 3) 62. f(x = 3 + In(2x)

63. 41-2x = 2 64. 86+3x = 4

In Problems 63-82, solve each equation. Express irrational solutions in exact form and as a decimal rounded to decimal places.

66. 4x-.1

67. log" 64 = -3 68. logyz x = -6

72. 252x = 5x'-12

78. log(7x - 12) = 210g x 79. el-x = 5

81. 9x + 4· 3x - 3 = 0

83. Suppose that f(x) = log2(x - 2) + 1.(a) Graph!(b) What isf(6)? What point is on the graph off?(c) Solve f(x) = 4. What point is on the graph off?(d) Based on the graph drawn in part (a), solve f(x) > O.(e) Find rl(x). Graph r: on the same Cartesian plane ss ].

12

70. 5x+: " y-2

73. log, Vx"=2 = 2 74.1'-1 8-x = 4

77. lo&.(x + 3) + gr,(x + 4) = 1

80. el-2x = 4

82. 4x - 14·4-x = 5

84. Suppose that f(x) = log3(x + 1) - 4(a) Graph [.(b) What is f(8)? What point is on th graph of f?(c) Solve f(x) = -3. What point is c the graph of f?(d) Based on the graph drawn in par: a), solve f(x) < O.(e) Find rl(x). Graph r1 on the sarr Cartesian plane as!

In Problems 85 and 86, use the following result: If x is the atmospheric pressure (measured in millimeters of mere ry}, then the formulafor the altitude h(x) (measured in meters above sea level) is

h(x) = (30T + 8000) 10g( :0)where T is the temperature (in degrees Celsius) and Po is the atmospheric pressure at sea level, which is approxiof mercury.

85. Finding the Altitude of an Airplane At what height is a PiperCub whose instruments record an outside temperature ofO°Cand a barometric pressure of 300 millimeters of mercury?

ately 760 millimeters

86. Finding thc Hl'ight of a :\)olllltain H w high is a mountainif instruments placed on its peak recor a temperature of 5°Cand a barometric pressure of 500 milli eters of mercury?

87. Amplifying Sound An amplifier's power output P (in watts)is related to its decibel voltage gain d by the formula

P = 25eo.ld.

(a) Find the power output for a de4 decibels.

(b) For a power output of 50 watts. W

age gain?

ibel voltage gain of

11 is the decibel volt-

88. .elescope is limited intar that it is aimed atsure of a star's bright-sr, the larger its mag-tude L of a telescope,

Limiting M:Jgnitlldc of a Tete-cope Pits usefulness by the brightness of theand by the diameter of its lens. One meness is its magnitude; the dimmer the ~nitude. A formula for the limiting magi

Chapter Review 421

ObjectivesSection You hould be able to: Review Exercises

6.1 Con .rt between decimals and degrees, minutes, seconds forms for angles (p. 346) 86

2 Find ile arc length of a circle (p. 347) 87,88

3 Con' -rt from degrees to radians and from radians to degrees (p. 348) 1-8

4 Find he area of a sector of a circle (p. 351) 87

5 Find he linear speed of an object traveling in circular motion (p. 352) 89-92

6.2 Find he exact values of the trigonometric functions using a point on the unitcircl- (p.359) 83,97

2 Find ne exact values of the trigonometric functions of quadrantal angles (p. 361) 10,17,18,20,97

3 Find le exact values of the trigonometric functions of ~ = 45° (p.363) 9,11,13,15,16

4 Find .ie exact values of the trigonometric functions of i = 30° and ~ = 60° (p.364) 9-15

5 Find ie exact values of the trigonometric functions for integer multiples of i = 30°,

tt 7r4

SO, and 3 = 60° (p. 366) 13-16,19,97

6 Use calculator to approximate the value of a trigonometric function (p. 367) 79,807 Use rcle of radius r to evaluate the trigonometric functions (p. 368) 84

6.3 Dete nine the domain and the range of the trigonometric functions (p. 373) 852 Dete nine the period of the trigonometric functions (p. 375) 853 Dete nine the signs of the trigonometric functions in a given quadrant (p. 377) 81,824 Find le values of the trigonometric functions using fundamental identities (p. 377) 21-305 Find re exact values of the trigonometric functions of an angle given

one, the functions and the quadrant of the angle (p. 380) 31-466 Use .en-odd properties to find the exact values of the trigonometric functions (p. 382) 27-30

6.4 Oral functions of the form y = A sin(wx) using transformations (p. 388) 472 Graj functions of the form y = A cos(wx) using transformations (p. 389) 483 Dete nine the amplitude and period of sinusoidal functions (p. 390) 63-684 Graj sinusoidal functions using key points (p. 392) 47,48,67,68,935 Find n equation for a sinusoidal graph (p.395) 75-78

6.5 Oral functions of the form y = A tan(wx) + Band y = A cot(wx) + B (p. 403) 53,54,562 Graj functions of the form y = A csc(wx) + Band y = A sec(wx) + B (p. 405) 57

6.6 Graj sinusoidal functions of the form y = A sine wx - cP) + B (p. 408) 49,50,59,60,69-74,942 Find sinusoidal function from data (p. 412) 95,96

Review Exercise ,

In Problems /-4, COf ert each angle in degrees to radians. Express your answer as a multiple of tr,

1. 135° 4. 15°

In Problems 5-8, COf

37r5'4

ert each angle in radians to degrees.

27r6'3

57r7. -2 37r

8. -2

In Problems 9-30, [ir ' the exact value of each expression. Do not use a calculator.

9. tan ~ - sin ~4 6

7r . 7r10. cos 3 + sm"2 11. 3 sin 45° - 4 tan i

12. 4 cos 60° + 3 ta -tt3

37r ( 7r)13. 6 cos 4 + 2 tan - 3 . 27r 57r14. 3 sm - - 4 cos -3 2

422 CHAPTER 6 Trigonometric Functions

31T (1T)16. 4csc4 - cot -4 17. tan 1T + sin 1T

18. cos~ - csc( -~) 19. cos 540° - tan( -405°) 20. sin 270° + cost -181-22. 2

cos 40°1 23. see 50° . cos 50°

24. tan 10°. cot 10° 25. sin 50°· csc 410°

27. sine-40°) . csc 40° 28. tan( -20°) . cot 200 29. sin 4050• sect -45") 30. cos 2500

• :c( -70°)

In Problems 31-46, find the exact value of each of the remaining trigonometric functions.

J . 4 1T 1 1T 12~.smO=5' 0<0<2 32.tanO=4,0<0<2 33.tanO=5' sinO 0

12 5 534. cot 0 = 5' cos 0 < 0 3.l..sec 0 = -- tan 0 < 0 I 36. csc 0 = - 3' cot 0 : 0. 4'

. 12 o in quadrant II3 . 5 31T

37. sm 0 = 13' 38 cos 0 = -- o in quadrant III 39. sin 0 = -13' - 0< 21T. 5' 2

12 31T 1 240. cos 0 = 13' T < 0 < 21T 41. tan 0 = 3' 180° < 0 < 270° 42. tan 0 = - 3' 90° . o < 180°

43. see 0 = 3,31T 31T 1T- < 0 < 21T 44. csc 0 = -4, 1T<O<T 45. cot 8 = -2, - <, <1T2 2

31T46. tan 0 = -2, T < 0 < 21T

In Problems 47-62, graph each function. Each graph should contain at least two periods.

47. y = 2sin(4x) 48. y = -3cos(2x) 49. y = -2COS(X +~) 50. y = 3 sin : - 1T)

51. Y = tan (x + 1T) 52. Y = -tan(x -~) 53. Y = -2 tan(3x) 54. Y = 4 tan 2x)

55. Y = cot( x + ~) 56. Y = -4 cot(2x) 57. Y = 4 sec(2x) 58. Y = csc( + ~)

62. y = 5 cot ~ - ~ )59. Y = 4 sin(2x + 4) - 2 60. y = 3 cos(4x + 2) + 1 61. Y = 4tan(1 +~)In Prohlems 63-66, determine the amplitude and period of each function without graphing.

65. y = -8 sin( ~ x )63. y = 4 cos x 64. y = sin(2x) 66. y = -2 C ;(31TX)

In Prohlems 67-74, find the amplitude, period, and phase shift of each function. Graph each function. Show at ( [Sf fWO periods.

67. y = 4 sin(3x) 68. y = 2 cos(~ x ) 69. y = 2 sin(2x - 1T) 70. Y = -CO'~X + ~)

1 . (3 )71. Y = "2 sin "2 x - 1T3

72. Y = "2 cos(6x + 31T)2

73 Y = --COS(1TX - 6). 3 74. Y = -7 s (~X +~)3 3

In Problems 75-78, 'nd a function whose graph is given.75. 76.

Chapter Review 423

77. 78.

79. Use a calculatetwo decimal pi

80. Use a calculateto two decimal

• • 1T"to approximate SIn 8"' Round the answer to.es,

to approximate see 10°. Round the answerlaces.

81. Determine the igns of the six trigonometric functions of anangle (J whose rminal side is in quadrant III.

82. Name the quae ant (J lies in if cas fJ > 0 and tan (J < O.

83. Find the exact rlues of the six trigonometric functions of t

if P = (- ~) :) is the point on the unit circle that3 .

corresponds to

84. Find the exact v ue of sin I, cas 1. and tan 1 if P = (-2,5) is the

point on the circ : that corresponds to I.

85. What is the do lain and the range of the secant function?What is the pel xi?

86. (a) Convert th angle 32°20'35" to a decimal in degrees.Round the nswer to two decimal places.

(b) Convert th angle 63.18° to DOM'S" form. Express theanswer to t ~ nearest second.

87. Find the length C the arc subtended by a central angle of 30°on a circle of ra. us 2 feet. What is the area of the sector?

88. The minute hat I of a clock is 8 inches long. How far doesthe tip of the I inute hand move in 30 minutes? How fardoes it move in 0 minutes?

89. Angular Speer! f a 1~:ll"eCar A race car is driven arounda circular tra " at a constant speed of 180 miles

per hour. If th diameter of the track is ~ mile, what is

the angular spe .I of the car? Express your answer in revo-lutions per hou: which is equivalent to laps per hour).

90. "Iern-Go-Holl .J.. A neighborhood carnival has a merry-go-round whose 'adius is 25 feet. If the time for one revolu-tion is 30 secom . how fast is the merry-go-round going?

91. Lighthouse He. 'ons The Montauk Point Lighthouse onLong Island has ual beams (two light sources opposite eachother). Ships at .a observe a blinking light every 5 seconds.What rotation s zed is required to do this?

92. Spin Halanl"ing irc, The radius of each wheel of a car is16 inches. At ho many revolutions per minute should a spin

balancer be set to balance the tires at a speed of 90 miles perhour? Is the setting different for a wheel of radius 14 inches?If so, what is this setting?

93. Alternating Voltagc The electromotive force E, in volts, ina certain ac (alternating circuit) circuit obeys the function

E(I) = 120 sin(1207Tt). t 2: 0

where t is measured in seconds.(a) What is the maximum value of E?(b) What is the period?(c) Graph this function over two periods.

94. Alternating Current The current I, in amperes, flowingthrough an ac (alternating current) circuit at time t is

let) = 220 sin(307Tf + i} 12:0

(a) What is the period?(b) What is the amplitude?(c) What is the phase shift?(d) Graph this function over two periods.

95. Monthly Temperature The following data represent the av-erage monthly temperatures for Phoenix, Arizona.

Average MonthlyMonth, m Temperature, T

January, 1 51

February, 2 55

March,3 63

April,4 67

May, 5 77

June,6 86

July, 7 90

August, 8 90

September, 9 84

October, 10 71

November, 11 59

December, 12 52

SOlJRr.[ U.S. National Oceanic and AtmosphericAdministration

424 CHAPTER 6 Trigonometric Functions

(a) Draw a scatter diagram of the data for one period.(b) Find a sinusoidal function of the form

y = A sin(wx - ¢) + B that fits the data.(c) Draw the sinusoidal function found in part (b) on the

scatter diagram.[-.j iell Use a graphing utility to find the sinusoidal function of...... best fit.

(,,) Graph the sinusoidal function of best fit on the scatterdiagram.

96. Hours of Daylight According to the Old Farmer's Almanac,in Las Vegas, Nevada, the number of hours of sunlight on thesummer solstice is 14.63 and the number of hours of sunlighton the winter solstice is 9.72.(a) Find a sinusoidal function of the form

y = A sine wx - ¢) + B that fits the data.(b) Use the function found in part (a) to predict the num-

ber of hours of sunlight on April 1, the 91st day of theyear.

(c) Draw a graph of the function found in part (a).(d) Look up the number of hours of sunlight for April 1 in

the Old Farmer's Almanac and compare the actual hoursof daylight to the results found in part (c).

97. Unit Circle On the unit circle below.gles (in radians) and the correspondin:each angle.

Angle:Y Angle:

pAngle:

Angle:

Angle:

Angle:

Angle:

Angle: Angle:

In Problems 1-3, convert each angle in degrees to radians. Expressyour answer as a multiple of 7T.

2. -400°

In Problems 4-fJ convert each angle in radians (0 degrees.

7T 97T 37T4. -8 5. 2 6. 4

In Problems 7-12, find the exact value of each expression.

(57T) 37T

8. cos -4 - cos410. tan 3300

. 7T7. SIn 69. cOS(_120D)

7T 197T11 sin - - tan--• 2 4 12. 2 sin2 60° - 3 cos 45°

In Problems 13-16, use a calculator to evaluate each expression.Round your answer to three decimal places.

27T14. cosS13. sin 17° 15. see 229°

287T16.cot -9-

17. Fill in each table entry with the sign of each function.

sin (} cos (} tan o sec (} csc (} cot (}

e in 01

e in 011

e in 011I

e in OIV

18. lff(x) = sin x andf(a) = ~,findf( a).

In Problems 19-2] find the value of the rimetric functions of (}.

19. sin () = %, () in quadrant II

2 37T20. COS() = -, - < ()< 27T

3 2

12 7T21. tan () = - - - < ()< 7T5 ' 2

In Problems 22-24, the point (x, y) is on t.gle () in standard position. Find the exatrigonometric function.

22. (2,7), sin ()

24. (6,-3),tan()

23. (-5,

In Problems 25 and 26, graph the function

2 2 . (x 7T)5. Y = SIn -;; - -.> 6

26. Y = (

.11in the missing an-:erminal points P of

19le:1-

Angle:f

Angle:i

Angle:x

Angle: 1~r.

Angle: 7;,ngle: 5;

wining five trigono-

. terminal side of an-value of the given

), cos ()

27. Write an equation for a sinusoidal gr ih with the followingproperties:

A = -3 . d 27Tpeno =:3 pha 'shift = - 7!.. 4

28. Logan has a g; den in the shape of a sector of a circle; theouter rim of th garden is 25 feet long and the central angleof the sector is iO°. She wants to add a 3-foot wide walk tothe outer rim; t w many square feet of paving blocks will sheneed to build t : walk?

29. Hungarian Ad an Annus won the gold medal for the ham-mer throw at t e 2004 Olympics in Athens with a winningdistance of 83. ) meters. * The event consists of swinging a

* Annus was strippec of his medal after refusing to cooperate withpostmedal drug testiru

CUMULATI\ : REVIEW13. Find a sinusoidal function for the following graph.1. Find the re: solutions, if any, of the equation

2X2 + x-I = J.

2. Find an equati. 1 for the line with slope -3 containing thepoint (-2,5).

3. Find an equati 1 for a circle of radius 4 and center at thepoint (0, -2).

4. Discuss the eql tion 2x - 3y = 12. Graph it.

5. Discuss the equ lion x2 + i -2x + 4y - 4 = O. Graph it.

6. Use transform, ons to graph the functiony = (x - 3)2 - 2.

7. Sketch a graphleast three poin(a) y = X2

(d) y = In x

8. Find the inversi

If each of the following functions. Label at; on each graph.

(b) Y = x3 ( C ) Y = eX

(e) y = sin x (f) y = tan xfunction of f(x) = 3x - 2.

lue of (sin 14°)2 + (cos 14°)2 - 3.

2x).

9. Find the exact,

10. Graph y = 3 si

7T 7T 7T11. Find the exact, lue of tan "4 - 3 cos"6 + csc"6'

12. Find an exponel ial function for the following graph. Expressyour answer in e form y = Ab",

y

____ .lo_

yo- 0 -6

(1,6)

2 4 6 x

Cumulative Review 425

16-pound weight attached to a wire 190 centimeters long-ina circle and then releasing it. Assuming his release is at a45° angle to the ground, the hammer will travel a distance ofv2-...!!. meters, where g = 9.8 meters/second/ and Vo is the lineargspeed of the hammer when released. At what rate (rpm) washe swinging the hammer upon release?

y

x

14. (a) Find a linear function that contains the points (-2,3)and (1, -6). What is the slope? What are the interceptsof the function? Graph the function. Be sure to label theintercepts.

(b) Find a quadratic function that contains the point(-2,3) with vertex (1, -6). What are the intercepts ofthe function? Graph the function.

(c) Show that there is no exponential function of the formf(x) = ae' that contains the points (-2,3) and (1, -6).

15. (a) Find a polynomial function of degree 3 whosey-intercept is 5 and whose x-intercepts are -2, 3, and 5.Graph the function.

(b) Find a rational function whose y-intercept is 5 andwhose x-intercepts are -2,3, and 5 that has the linex = 2 as a vertical asymptote. Graph the function.

426 CHAPTER 6 Trigonometric Functions

CHAPTER PROJECTS

3. On your graphing utility, draw a sc:data in the table. Let [ (time) be table. with [ = 0 being 12:00 AM onbeing 12:00 AM on November 15. anheight in feet. Remember that therehour. Also. make sure your graphiimode.

4. What shape does the data take? Whdata? What is the amplitude? Is theExplain.

5. Fit a sine curve to the data. Is thethere a phase shift?

6. Using your graphing utility. find trof best fit. How does this functiiequation?

7. Using the equation found in part (equation of best fit found in part (6)and the low tides on November 21.

8. Looking at the times of day that thedo you think causes the low tides I

day? Explain. Does this seem to heffect on the high tides? Explain.

I. Tides The given table is a partial tide table for November2006 for the Sabine Bank Lighthouse, a shoal located off-shore from Texas where the Sabine River empties into theGulf of Mexico.

1. On November 15. when was the tide high? This is calledhigh [ide. On November 19, when was the tide low? Thisis called low [ide. Most days will have two high tides andtwo low tides.

2. Why do you think there is a negative height for the low tideon November 20? What is the height measured against?

Low Tide Low Tide High Tide High Tide Sun/M

Noy Time Ht (ft) Time Ht (ft) Time Ht (ft) Time Ht (ft) Sunrise/set

14 6:26a 2.0 4:38p 1.4 9:29a 2.2 11:14p 2.8 6:40a/S:20p

15 6:22a 1.6 S:34p 1.8 11:18a 2.4 11:lSp 2.6 6:41 a/S:20p

16 6:28a 1.2 6:2Sp 2.0 12:37p 2.6 11:16p 2.6 6:41 a/S:19p

17 6:40a 0.8 7:12p 2.4 1:38p 2.8 11:16p 2.6 6:42a/S:19p

18 6:S6a 0.4 7:S7p 2.6 2:27p 3.0 11:14p 2.8 6:43a/S:19p

19 7:17a 0.0 8:38p 2.6 3:10p 3.2 11:0Sp 2.8 6:44a/S:18p

20 7:43a -0.2 3:S2p 3.4 6:4Sa/S:18p

[Note: a, AM: p, PM.]

Sources: National Oceanic and Atmospheric Administration (hllp.//tidesandcurrenls.noaa.gov) and u.s. Naval Observa. I'V

(hnpi//aa. usno.navy. m ill

The following projects are available at the Instructor's Resource Center (IRC):

ter diagram for the~ independent vari-overnber 14. [ = 24so on. Let h be the.re 60 minutes in an

utility is in radian

is the period of theimplitude constant?

a vertical shift? Is

sinusoidal functionI compare to your

) and the sinusoidaliredict the high tides

ow tides occur. whatvary so much each

/e the same type of

an Phase

Moonrise/set

1 :OSa/2:02p

1:S8a/2:27p

2:S0a/2:S2p

3:43a/3:19p

4:38a/3:47p

S:3Sa/4:20p

6:34a/4:S7p

II. Project at Motorola Digital Transmission over the Air Learn how Motorola Corporation transmits digit I sequences by mod-ulating the phase of the carrier waves.

III. Identifying Mountain Peaks in Hawaii The visibility of a mountain is affected by its altitude. distance fro I the viewer. and thecurvature of the earth's surface. Trigonometry can be used to determine whether a distant object can be sc n.

IV. CBl Experiment Technology is used to model and study the effects of damping on sound waves.