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Page 1 of 2 TRIGONOMETRIC GRAPHS, IDENTITIES, AND EQUATIONS How long does it take to go around a Ferris wheel? 828 Page 2 of 2 CHAPTER 14 APPLICATION: Ferris Wheels The Texas State Fair is home to America’s tallest Ferris wheel. The Ferris wheel takes riders to a height of 212 feet and has a diameter of about 203 feet. The maximum speed of the Ferris wheel is 1.5 rotations per minute. Think & Discuss Height (feet) The graph below shows a person’s height (in feet) above the ground while riding the Ferris wheel at maximum speed. Use the graph to answer the questions below. y 200 150 100 50 0 0 20 40 60 80 100 x Time (seconds) 1. How many seconds does it take to reach the maximum height from the minimum height? 2. How long does it take to complete one revolution? 3. How would the graph change if the Ferris wheel rotated faster? if the Ferris wheel had a smaller diameter? Learn More About It INT In Example 5 on p. 842 you will use trigonometric functions to model a person’s height above the ground while riding a Ferris wheel. NE ER T APPLICATION LINK Visit www.mcdougallittell.com for more information about Ferris wheels. 829 Page 1 of 1 Study Guide CHAPTER 14 PREVIEW What’s the chapter about? Chapter 14 is about trigonometry. In Chapter 14 you’ll learn • • • how to graph trigonometric functions and transformations of trigonometric graphs. how to use trigonometric identities and solve trigonometric equations. how to write and use trigonometric models. K E Y VO C A B U L A RY Review • identity, p. 13 • domain, p. 67 • range, p. 67 • x-intercept, p. 84 • quadratic form, p. 346 PREPARE • local maximum, p. 374 • local minimum, p. 374 • asymptote, p. 465 • sine, p. 769 • cosine, p. 769 • tangent, p. 769 New • periodic function, p. 831 • cycle, p. 831 • period, p. 831 • amplitude, p. 831 • trigonometric identities, p. 848 Are you ready for the chapter? SKILL REVIEW Do these exercises to review key skills that you’ll apply in this chapter. See the given reference page if there is something you don’t understand. Graph the function. (Review Example 1, p. 123; Example 2, p. 250; Example 2, p. 624) STUDENT HELP Study Tip “Student Help” boxes throughout the chapter give you study tips and tell you where to look for extra help in this book and on the Internet. 1. y = º|x + 2| º 4 2. y = 2(x + 2)2 + 3 3. (x + 4)2 + (y º 1)2 = 9 Solve the equation. (Review Example 5, p. 258; Example 1, p. 291) 4. x 2 + 7x º 8 = 0 5. 9x 2 º 25 = 0 6. 3x 2 º x º 5 = 0 Evaluate the function without using a calculator. (Review Example 4, p. 786) π 8. tan 30° 9. cos 10. sin π 7. sin 60° 4 Evaluate the expression without using a calculator. Give your answer in both radians and degrees. (Review Example 1, p. 793) 3 2 11. sinº1 12. cosº1 13. cosº1 0 14. tanº1 (º3 ) 2 2 STUDY STRATEGY 830 Chapter 14 Here’s a study strategy! Multiple Methods There is often more than one wa Multiple methods can help in y to do an exercise. three situations. (1) If you get stuck using one me (2) Do an exercise more than thod, try another. one your understanding. (3) Check way to reinforce your work by using a different method. Page 1 of 7 14.1 What you should learn GOAL 1 Graph sine and cosine functions, as applied in Example 3. GOAL 2 Graphing Sine, Cosine, and Tangent Functions GOAL 1 In this lesson you will learn to graph functions of the form y = a sin bx and y = a cos bx where a and b are positive constants and x is in radian measure. The graphs of all sine and cosine functions are related to the graphs of Graph tangent y = sin x functions. Why you should learn it and y = cos x which are shown below. y M1 1 amplitude: 1 range: 1 ≤ y ≤ 1 FE To model repeating reallife patterns, such as the vibrations of a tuning fork in Ex. 52. AL LI RE GRAPHING SINE AND COSINE FUNCTIONS π 3π 2 π π π 2 2 m 1 2π x 3π 2 period: 2π Graph of y = sin x y M1 range: 1 ≤ y ≤ 1 amplitude: 1 2π 3π 2 π m 1 π 2 π 2 1 π 3π 2 2π x period: 2π Graph of y = cos x The functions y = sin x and y = cos x have the following characteristics. 1. The domain of each function is all real numbers. 2. The range of each function is º1 ≤ y ≤ 1. 3. Each function is periodic, which means that its graph has a repeating pattern that continues indefinitely. The shortest repeating portion is called a cycle. The horizontal length of each cycle is called the period. Each graph shown above has a period of 2π. π 4. The maximum value of y = sin x is M = 1 and occurs when x = + 2nπ 2 where n is any integer. The maximum value of y = cos x is also M = 1 and occurs when x = 2nπ where n is any integer. 3π 5. The minimum value of y = sin x is m = º1 and occurs when x = + 2nπ 2 where n is any integer. The minimum value of y = cos x is also m = º1 and occurs when x = (2n + 1)π where n is any integer. 1 6. The amplitude of each function’s graph is (M º m) = 1. 2 14.1 Graphing Sine, Cosine, and Tangent Functions 831 Page 2 of 7 CHARACTERISTICS OF Y = A SIN BX AND Y = A COS BX The amplitude and period of the graphs of y = a sin bx and y = a cos bx, where a and b are nonzero real numbers, are as follows: amplitude = |a| Examples 2π period = and |b| 2π 4 π 2 The graph of y = 2 sin 4x has amplitude 2 and period = . 1 3 1 3 2π 2π The graph of y = cos 2πx has amplitude and period = 1. For a > 0 and b > 0, the graphs of y = a sin bx and y = a cos bx each have five 2π b key x-values on the interval 0 ≤ x ≤ : the x-values at which the maximum and minimum values occur and the x-intercepts. y (0, 0) 14 p 2πb , a 12 p 2πb , 0 EXAMPLE 1 y = a sin bx y = a cos bx y (0, a) 2πb , 0 2πb , a 14 p 2πb , 0 x 34 p 2πb , a 12 p 2πb , a 34 p 2πb , 0 x Graphing Sine and Cosine Functions Graph the function. a. y = 2 sin x b. y = cos 2x SOLUTION 2π 2π a. The amplitude is a = 2 and the period is = = 2π. The five key points are: b 1 Intercepts: (0, 0); (2π, 0); 14 • 2π, 2 = π2, 2 34 • 2π, º2 = 32π, º2 1 • 2π, 0 = (π, 0) 2 Maximum: STUDENT HELP Study Tip In Example 1 notice how changes in a and b affect the graphs of y = a sin bx and y = a cos bx. When the value of a increases, the amplitude is greater. When the value of b increases, the period is shorter. 832 Minimum: y 1 π 2 3π 2 x 2π 2π b. The amplitude is a = 1 and the period is = = π. The five key points are: b 2 Intercepts: 14 • π, 0 = π4, 0; 34 • π, 0 = 34π, 0 Maximums: (0, 1); (π, 1) Minimum: 12 • π, º1 = π2, º1 Chapter 14 Trigonometric Graphs, Identities, and Equations y 2 π x Page 3 of 7 EXAMPLE 2 Graphing a Cosine Function 1 3 Graph y = cos πx. SOLUTION 1 3 2π b 2π π The amplitude is a = and the period is = = 2. The five key points are: Intercepts: Maximums: Minimum: 14 • 2, 0 = 12, 0; 34 • 2, 0 = 32, 0 0, 13; 2, 13 12 • 2, º13 = 1, º13 y 2 3 2 x .......... The periodic nature of trigonometric functions is useful for modeling oscillating motions or repeating patterns that occur in real life. Some examples are sound waves, the motion of a pendulum or a spring, and seasons of the year. In such applications, the reciprocal of the period is called the frequency. The frequency gives the number of cycles per unit of time. EXAMPLE 3 Modeling with a Sine Function MUSIC When you strike a tuning fork, the vibrations cause changes in the pressure of the surrounding air. A middle-A tuning fork vibrates with frequency f = 440 hertz (cycles per second). You strike a middle-A tuning fork with a force that produces a maximum pressure of 5 pascals. a. Write a sine model that gives the pressure P as a function of time t (in seconds). b. Graph the model. FOCUS ON APPLICATIONS SOLUTION a. In the model P = a sin bt, the maximum pressure P is 5, so a = 5. You can use the frequency to find the value of b. 1 period b 2π frequency = 440 = 880π = b The pressure as a function of time is given by P = 5 sin 880πt. 1 1 b. The amplitude is a = 5 and the period is = . ƒ 440 The five key points are: RE FE L AL I 4140 12 • 4140, 0 = 8180, 0 14 • 4140, 5 = 17160, 5 34 • 4140, º5 = 17360, º5 Intercepts: (0, 0); , 0 ; OSCILLOSCOPE The oscilloscope is a laboratory device invented by Karl Braun in 1897. This electrical instrument measures waveforms and is the forerunner of today’s television. Maximum: Minimum: P 5 1 880 1 440 14.1 Graphing Sine, Cosine, and Tangent Functions t 833 Page 4 of 7 GOAL 2 GRAPHING TANGENT FUNCTIONS The graph of y = tan x has the following characteristics. π π 1. The domain is all real numbers except odd multiples of . At odd multiples of , 2 2 the graph has vertical asymptotes. 2. The range is all real numbers. 3. The graph has a period of π. C H A R A C T E R I S T I C S O F Y = A TA N B X If a and b are nonzero real numbers, the graph of y = a tan bx has these characteristics: π |b| • The period is . • There are vertical asymptotes at odd multiples of . π 2|b| π 3 The graph of y = 5 tan 3x has period and asymptotes at Example π 2(3) π 6 nπ 3 x = (2n + 1) = + where n is any integer. The graph at the right shows five key x-values that can help you sketch the graph of y = a tan bx for a > 0 and b > 0. These are the x-intercept, the x-values where the asymptotes occur, and the x-values halfway between the x-intercept and the asymptotes. At each halfway point, the function’s value is either a or ºa. y a π 2b π 4b π 2b x Graphing a Tangent Function EXAMPLE 4 3 2 Graph y = tan 4x. SOLUTION π b π 4 The period is = . y Intercept: (0, 0) Asymptotes: x = • , or x = ; 1 π 2 4 1 π π x = º • , or x = º 2 4 8 14 • π4, 32 = 1π6, 32; º14 • π4, º32 = º1π6 , º32 Halfway points: 834 1 π 8 Chapter 14 Trigonometric Graphs, Identities, and Equations π8 π 16 π 8 x Page 5 of 7 GUIDED PRACTICE ✓ Concept Check ✓ Vocabulary Check 1. Define the terms cycle and period. 2. What are the domain and range of y = a sin bx, y = a cos bx, and y = a tan bx? 1 x 3. Consider the two functions y = 4 sin and y = sin 4x. Which function has the 3 3 greater amplitude? Which function has the longer period? Skill Check ✓ Find the amplitude and period of the function. 4. y = 6 sin x 5. y = 3 cos πx 2 π 7. y = sin x 3 3 8. y = 5 sin 3πx 1 6. y = cos 3x 4 x 9. y = cos 2 Graph the function. 10. y = 3 sin x 11. y = cos 4x 12. y = tan 3x 1 13. y = sin πx 4 2 14. y = 5 cos x 3 15. y = 2 tan 4x 16. PENDULUMS The motion of a certain pendulum can be modeled by the function d = 4 cos 8πt where d is the pendulum’s horizontal displacement (in inches) relative to its position at rest and t is the time (in seconds). Graph the function. How far horizontally does the pendulum travel from its original position? d PRACTICE AND APPLICATIONS STUDENT HELP MATCHING GRAPHS Match the function with its graph. Extra Practice to help you master skills is on p. 959. 1 17. y = 2 sin x 2 1 18. y = 2 cos x 2 19. y = 2 sin 2x 1 20. y = 2 tan x 2 21. y = 2 cos 2x 22. y = 2 tan 2x A. y B. 2 C. y 1 x y 1 x π π π4 3π π 4 x π x STUDENT HELP HOMEWORK HELP Examples 1, 2: Exs. 17–49 Example 3: Exs. 51–55 Example 4: Exs. 17–22, 32–43 D. E. y 1 x π F. y y 1 1 π 2π x π 14.1 Graphing Sine, Cosine, and Tangent Functions 835 Page 6 of 7 INT STUDENT HELP NE ER T HOMEWORK HELP Visit our Web site www.mcdougallittell.com for help with problem solving in Exs. 44–49. ANALYZING FUNCTIONS In Exercises 23–31, find the amplitude and period of the graph of the function. 23. 24. y 25. y y 0.5 1 π 1 26. y = cos πx 2 1 29. y = 5 cos x 2 x 1 2 x 27. y = sin 2x 1 30. y = 2 sin πx 2 2 π 2 π 1 28. y = 3 cos x 4 1 31. y = sin 4πx 3 GRAPHING Draw one cycle of the function’s graph. 1 32. y = sin x 4 1 33. y = cos x 5 1 34. y = tan πx 4 1 35. y = sin x 4 36. y = 4 cos x 37. y = 4 tan 2x 38. y = 3 cos 2x 39. y = 8 sin x 1 40. y = 2 tan x 3 1 1 41. y = sin πx 2 4 42. y = tan 4πx 43. y = 2 cos 6πx WRITING EQUATIONS Write an equation of the form y = a sin bx, where a > 0 and b > 0, so that the graph has the given amplitude and period. 44. Amplitude: 1 Period: 5 1 47. Amplitude: 2 Period: 3π FOCUS ON CAREERS RE FE L AL I INT NE ER T CAREER LINK www.mcdougallittell.com 836 46. Amplitude: 2 Period: 4 48. Amplitude: 4 π Period: 6 Period: 2π 49. Amplitude: 3 1 Period: 2 50. LOGICAL REASONING Use the fact that the frequency of a periodic function’s graph is the reciprocal of the period to show that an oscillating motion with maximum displacement a and frequency ƒ can be modeled by y = a sin 2π ft or y = a cos 2π ft. 51. BOATING The displacement d (in feet) of a boat’s water line above sea level as it moves over waves can be modeled by the function d = 2 sin 2πt where t is the time (in seconds). Graph the height of the boat over a three second time interval. 52. MUSIC A tuning fork vibrates with a frequency of 220 hertz (cycles per second). You strike the tuning fork with a force that produces a maximum pressure of 3 pascals. Write a sine model that gives the pressure P as a function of the time t (in seconds). What is the period of the sound wave? 53. SPRING MOTION The motion of a simple spring can be modeled by y = A cos kt where y is the spring’s vertical displacement (in feet) relative to its position at rest, A is the initial displacement (in feet), k is a constant that measures the elasticity of the spring, and t is the time (in seconds). Find the amplitude and period of a spring for which A = 0.5 foot and k = 6. MUSICIAN Musicians may specialize in classical, rock, jazz, or many other types of music. This profession is very competitive and demands a high degree of discipline and talent in order to succeed. 45. Amplitude: 10 Chapter 14 Trigonometric Graphs, Identities, and Equations A A x Page 7 of 7 SIGHTSEEING In Exercises 54 and 55, use the following information. Suppose you are standing 100 feet away from the base Not drawn of the Statue of Liberty with a video camera. As you to scale videotape the statue, you pan up the side of the statue at 5° per second. 54. Write and graph an equation that gives the height you h of the part of the statue seen through the video camera as a function of the time t. h † 100 ft 55. Find the change in height from t = 0 to t = 1, from t = 1 to t = 2, and from t = 2 to t = 3. Briefly explain what happens to h as t increases. Test Preparation 56. MULTIPLE CHOICE Which function represents the graph shown? A ¡ C ¡ E ¡ y = tan 5x 1 2 B ¡ y = 5 tan x y = tan 5x D ¡ y = 5 tan 2x 1 2 y 5 π2 , 5 π π x π 2 1 y = 5 tan x 8 57. MULTIPLE CHOICE Which of the following is an x-intercept of the graph of 1 π y = sin x? 3 4 ★ Challenge A ¡ 4 B ¡ C ¡ 2 º6 D ¡ 1 E ¡ 4π SKETCHING GRAPHS Sketch the graph of the function by plotting points. Then state the function’s domain, range, and period. 58. y = csc x 59. y = sec x 60. y = cot x MIXED REVIEW GRAPHING Graph the quadratic function. Label the vertex and axis of symmetry. (Review 5.1 for 14.2) 61. y = 2(x º 5)2 + 4 62. y = º(x º 3)2 º 7 63. y = 4(x + 2)2 º 1 64. y = º3(x + 1)2 + 6 3 65. y = (x º 1)2 º 2 4 66. y = 10(x + 4)2 + 3 CALCULATING PROBABILITY Find the probability of drawing the given numbers if the integers 1 through 30 are placed in a hat and drawn randomly without replacement. (Review 12.5) 67. an even number, then an odd number 68. the number 30, then an odd number 69. a multiple of 4, then an odd number 70. the number 19, then the number 20 FINDING REFERENCE ANGLES Sketch the angle. Then find its reference angle. (Review 13.3) 71. 220° 72. º155° 73. 280° 74. º510° 35π 21π 17π 5π 75. 76. 77. º 78. º 3 4 6 8 79. PERSONAL FINANCE You deposit $1000 in an account that pays 1.5% annual interest compounded continuously. How long will it take for the balance to double? (Review 8.6) 14.1 Graphing Sine, Cosine, and Tangent Functions 837 Page 1 of 1 ACTIVITY 14.1 Using Technology Graphing Calculator Activity for use with Lesson 14.1 Graphing Trigonometric Functions Using the List feature and the trigonometric viewing window of a graphing calculator, you can graph the trigonometric functions y = a sin bx, y = a cos bx, and y = a tan bx and observe the effects of changing the values of a and b. EXAMPLE Graph y = a sin x for a = 1, 2, and 4. SOLUTION 1 Use the graphing calculator’s List feature to INT STUDENT HELP NE ER T KEYSTROKE HELP See keystrokes for several models of calculators at www.mcdougallittell.com enter the function as y1 = {1, 2, 4} sin x. This represents the three functions y = sin x, y = 2 sin x, and y = 4 sin x. 2 Select the trigonometric viewing window. In this π window, each tick mark on the x-axis represents 2 and each tick mark on the y-axis represents 1. Y1={1,2,4}sin(X) Y2 = Y3 = Y4 = Y5 = Y6 = Y7 = ZOOM MEMORY 1:ZBox 2:Zoom In 3:Zoom Out 4:ZDecimal 5:ZSquare 6:ZStandard 7 ZTrig 3 If the functions you graph have amplitudes much greater than 1 or periods longer than 4π, you may need to change the parameters of the viewing window. Then you can use the Maximum, Minimum, and Zero features to find the amplitude and the period of each function. EXERCISES Use a graphing calculator’s List feature to graph the functions for the given values of a in the same trigonometric viewing window. Find the amplitude and the period of each function’s graph. 1 1. y = a sin x for a = , 3, 9 2. y = a sin x for a = 10, 20, 40 3 1 3. y = a cos x for a = , 1, 2 4. y = a cos x for a = 1, 2, 4 2 Use a graphing calculator’s List feature to graph the functions for the given values of b in the same trigonometric viewing window. Find the amplitude and the period of each function’s graph. 1 5. y = sin bx for b = 1, 2, 4 6. y = sin bx for b = , 1, 3 3 1 1 7. y = cos bx for b = , 1, 2 8. y = cos bx for b = , 1, 3 2 3 838 Chapter 14 Trigonometric Graphs, Identities, and Equations Page 1 of 1 ACTIVITY 14.2 Developing Concepts GROUP ACTIVITY Group Activity for use with Lesson 14.2 Translating and Reflecting Trigonometric Graphs QUESTION How can you graph reflections and horizontal and vertical translations of graphs of trigonometric functions? Work with a partner. MATERIALS graphing calculator EXPLORING THE CONCEPT 1 Use a graphing calculator to graph each pair of functions in the same viewing window. How are the graphs geometrically related? a. y = cos x 3 c. y = sin 3x 5 3 y = º sin 3x 5 b. y = 2 sin x y = ºcos x y = º2 sin x 2 Use a graphing calculator to graph each pair of functions in the same viewing window. How are the graphs geometrically related? a. y = cos x b. y = 2 sin x y = cos x º π 2 c. y = sin 3x y = 2 sin x + π 2 y = sin 3(x + π) 3 Use a graphing calculator to graph each pair of functions in the same viewing window. How are the graphs geometrically related? a. y = cos x 1 c. y = sin 3x 4 1 y = 4 sin 3x º 2 b. y = 2 sin x y = cos x + 1 y = 2 sin x º 1 DRAWING CONCLUSIONS 1. Predict what the graph of each function looks like by making a sketch. Check your prediction by graphing the function on a graphing calculator. a. y = º3 cos x 3π b. y = sin x + 4 c. y = sin x º 4 2. Predict what the graph of each function looks like by making a sketch. Check your prediction by graphing the function on a graphing calculator. 1 a. y = º cos x º 2 2 b. y = º3 cos (x + π) c. y = 2 sin (x º π) + 3 π d. y = º4 sin x º º 6 4 3. CRITICAL THINKING Use the following phrases to describe how the graph of each function in parts (a)–(d) is related to the graph of y = sin x. • • • shifted up 1 unit shifted left 1 unit • • shifted down 1 unit shifted right 1 unit reflected in a horizontal line a. y = sin (x º 1) + 1 b. y = ºsin (x º 1) º 1 c. y = ºsin (x + 1) + 1 d. y = sin (x + 1) º 1 14.2 Concept Activity 839 Page 1 of 8 14.2 What you should learn GOAL 1 Graph translations and reflections of sine and cosine graphs. Translations and Reflections of Trigonometric Graphs GOAL 1 GRAPHING SINE AND COSINE FUNCTIONS In previous chapters you learned that the graph of y = a • ƒ(x º h) + k is related to the graph of y = |a| • ƒ(x) by horizontal and vertical translations and by a reflection when a is negative. This also applies to sine, cosine, and tangent functions. GOAL 2 Graph translations and reflections of tangent graphs, as applied in Ex. 61. T R A N S F O R M AT I O N S O F S I N E A N D C O S I N E G R A P H S Why you should learn it To obtain the graph of RE FE To model real-life quantities, such as the height above the ground of a person rappelling down a cliff in Example 7. AL LI y = a sin b(x º h) + k or y = a cos b(x º h) + k, transform the graph of y = |a| sin bx or y = |a| cos bx as follows. VERTICAL SHIFT Shift the graph k units vertically. HORIZONTAL SHIFT Shift the graph h units horizontally. REFLECTION If a < 0, reflect the graph in the line y = k after any vertical and horizontal shifts have been performed. Graphing a Vertical Translation EXAMPLE 1 Graph y = º2 + 3 sin 4x. SOLUTION Because the graph is a transformation of the graph of y = 3 sin 4x, the amplitude is 3 2π π and the period is = . By comparing the given equation to the general equation 4 2 y = a sin b(x º h) + k, you can see that h = 0, k = º2, and a > 0. Therefore, translate the graph of y = 3 sin 4x down 2 units. The graph oscillates 3 units up and down from its center line y = º2. Therefore, the maximum value of the function is º2 + 3 = 1 and the minimum value of the function is º2 º 3 = º5. y 1 π 8 π 4 3π 8 π 2 The five key points are: π4 π2 On y = k : (0, º2); , º2 ; , º2 π8 38π, º5 Maximum: , 1 Minimum: ✓CHECK You can check your graph with a graphing calculator. Use the Maximum, Minimum, and Intersect features to check the key points. 840 Chapter 14 Trigonometric Graphs, Identities, and Equations x Page 2 of 8 Graphing a Horizontal Translation EXAMPLE 2 Graph y = 2 cos 2 x º π . 3 4 SOLUTION Because the graph is a transformation of the graph of y = 2 cos 2x, the amplitude is 2 3 2π = 3π. By comparing the given equation to the general equation and the period is 2 3 y = a cos b(x º h) + k, you can see that h = π, k = 0, and a > 0. Therefore, 4 translate the graph of y = 2 cos 2x right π unit. (Notice that the maximum occurs 3 π unit to the right of the y-axis.) 4 4 The five key points are: STUDENT HELP Study Tip When graphing translations of functions, you may find it helpful to graph the basic function first and then translate the graph. On y = k : Maximums: Minimum: 14 • 3π + π4, 0 = (π, 0); 34 • 3π + π4, 0 = 52π, 0 0 + π4, 2 = π4, 2; 3π + π4, 2 = 134π , 2 12 • 3π + π4, º2 = 74π, º2 EXAMPLE 3 y 3 π 2π 3π x Graphing a Reflection Graph y = º3 sin x. SOLUTION Because the graph is a reflection of the graph of y = 3 sin x, the amplitude is 3 and the period is 2π. When you plot the five key points on the graph, note that the intercepts are the same as they are for the graph of y = 3 sin x. However, when the graph is reflected in the x-axis, the maximum becomes a minimum and the minimum becomes a maximum. The five key points are: On y = k : (0, 0); (2π, 0); Minimum: Maximum: 12 • 2π, 0 = (π, 0) 14 • 2π, º3 = π2, º3 34 • 2π, 3 = 32π, 3 y 3 π 2 π 3π 2 2π x .......... The next example shows how to graph a function when multiple transformations are involved. 14.2 Translations and Reflections of Trigonometric Graphs 841 Page 3 of 8 EXAMPLE 4 Combining a Translation and a Reflection 1 2 Graph y = º cos (2x + 3π) + 1. SOLUTION Begin by rewriting the function in the form y = a cos b(x º h) + k: 1 2 1 3π +1 2 2 1 2π 3π The amplitude is and the period is = π. Since h = º, k = 1, and a < 0, the 2 2 2 1 3π graph of y = cos 2x is shifted left units and up 1 unit, and then reflected in the 2 2 y = º cos (2x + 3π) + 1 = º cos 2 x º º line y = 1. The five key points are: On y = k : Minimums: Maximum: EXAMPLE 5 14 • π º 32π, 1 = º54π, 1; 34 • π º 32π, 1 = º34π, 1 0 º 32π, 1 º 12 = º32π, 12; π º 32π, 1 º 12 = ºπ2, 12 12 • π º 32π, 1 + 12 = ºπ, 32 y 2 3π 2 π π 2 x Modeling Circular Motion FERRIS WHEEL You are riding a Ferris wheel. Your height h (in feet) above the ground at any time t (in seconds) can be modeled by the following equation: π 15 h = 25 sin t º 7.5 + 30 The Ferris wheel turns for 135 seconds before it stops to let the first passengers off. FOCUS ON APPLICATIONS a. Graph your height above the ground as a function of time. b. What are your minimum and maximum heights above the ground? SOLUTION 135 2π = 30. The wheel turns = 4.5 times a. The amplitude is 25 and the period is π 30 15 in 135 seconds, so the graph shows 4.5 cycles. The five key points are (7.5, 30), (15, 55), (22.5, 30), (30, 5), and (37.5, 30). h RE FE L AL I THE FERRIS WHEEL was invented in 1893 by George Washington Gale Ferris for the World’s Columbian Exposition in Chicago. Each car held 60 people. NE ER T 50 10 t b. Since the amplitude is 25 and the graph is shifted up 30 units, the maximum height is 30 + 25 = 55 feet and the minimum height is 30 º 25 = 5 feet. INT APPLICATION LINK www.mcdougallittell.com 842 Chapter 14 Trigonometric Graphs, Identities, and Equations Page 4 of 8 GOAL 2 GRAPHING TANGENT FUNCTIONS Graphing tangent functions using translations and reflections is similar to graphing sine and cosine functions. T R A N S F O R M AT I O N S O F TA N G E N T G R A P H S To obtain the graph of y = a tan b(x º h) + k, transform the graph of y = |a| tan bx as follows. • • Shift the graph k units vertically and h units horizontally. Then, if a < 0, reflect the graph in the line y = k. EXAMPLE 6 Combining a Translation and a Reflection π 4 Graph y = º2 tan x + . SOLUTION The graph is a transformation of the graph of y = 2 tan x, so the period is π. By comparing the given equation to y π 4 y = a tan b(x º h) + k, you can see that h = º, k = 0, 1 and a < 0. Therefore, translate the graph of y = 2 tan x π 4 left unit and then reflect it in the x-axis. 4 3 4 π 3π π π π π 4 4 4 4 2•1 2•1 π On y = k : (h, k) = º, 0 4 π π π π π Halfway points: º º , 2 = º, 2 ; º , º2 = (0, º2) 4 2 4 4•1 4•1 x Asymptotes: x = º º = º; x = º = Rappelling EXAMPLE 7 Modeling with a Tangent Function You are standing 200 feet from the base of a 180 foot cliff. Your friend is rappelling down the cliff. Write and graph a model for your friend’s distance d from the top as a function of her angle of elevation †. Not drawn to scale your friend d 180 d you † 200 ft SOLUTION Use the tangent function to write an equation relating d and †. op p ad j 180 º d 2 00 tan † = = Definition of tangent 200 tan † = 180 º d Multiply each side by 200. d = º200 tan † + 180 Solve for d. Distance from top (ft) RE FE L AL I d 180 120 60 0 0 20 40 † 60 Angle (degrees) 14.2 Translations and Reflections of Trigonometric Graphs 843 Page 5 of 8 GUIDED PRACTICE ✓ Concept Check ✓ Vocabulary Check ? shifts a graph horizontally or vertically. 1. Complete this statement: A(n) 2. How is the graph of y = º2 cos 3x related to the graph of y = 2 cos 3x? 3. How is the graph of y = tan 2(x º π) related to the graph of y = tan 2x? Skill Check ✓ State whether the graph of the function is a vertical shift, a horizontal shift, and/or a reflection of the graph of y = 4 cos 2x. 4. y = 3 + 4 cos 2x 5. y = 4 cos (2x + 1) 6. y = º4 cos 2x 7. y = 4 cos 2(x + 1) 8. y = 4 cos (2x º 1) + 3 9. y = º3 º 4 cos 2x Graph the function. 10. y = 3 sin (x + π) 11. y = º2 cos x + 1 13. y = ºsin π(x º 2) + 3 14. y = 4 cos 2(x º π) + 1 15. y = 5 º tan 2(x º π) 16. 12. y = 2 tan (x º π) RAPPELLING Look back at Example 7. Suppose the cliff is 250 feet high and you are 150 feet from the base. Write and graph an equation that gives your friend’s distance from the top as a function of her angle of elevation. PRACTICE AND APPLICATIONS STUDENT HELP Extra Practice to help you master skills is on p. 959. TRANSFORMING GRAPHS Describe how the graph of y = sin x or y = cos x can be transformed to produce the graph of the given function. 17. y = 2 + sin x π 20. y = cos x + 2 π 23. y = 5 º cos x º 4 18. y = 5 º cos x 19. y = º2 + cos x 21. y = ºsin (x + π) π 22. y = sin x º 2 24. y = º2 º sin (x º π) 3π 25. y = 3 + cos x + 4 MATCHING Match the function with its graph. 26. y = º2 + sin (2x + π) 27. y = ºsin (x + π) π 29. y = cos x + 2 1 30. y = 1 + sin x 2 y A. 28. y = º3 + cos x 3131..y = 1 + 2 cos y B. C. 12x + π2 y 2 1 1 x π π 2 2 STUDENT HELP HOMEWORK HELP Examples 1–4: Exs. 17–55 Example 5: Exs. 57–60 Example 6: Exs. 32–55 Example 7: Exs. 61, 62 844 y D. y E. 3 1 π x x Chapter 14 Trigonometric Graphs, Identities, and Equations π x π 2 x y F. π x 3 Page 6 of 8 GRAPHING Graph the function. INT NE ER T HOMEWORK HELP Visit our Web site www.mcdougallittell.com for help with Exs. 50–55. π 33. y = cos x º 2 1 34. y = º4 sin x 4 35. y = 1 + cos (x + π) 36. y = º1 + cos (x º π) 37. y = 2 + 3 sin (4x + π) 38. y = º4 + sin 2x 39. y = 1 + 5 cos (x º π) 40. y = 2 º sin x 3π 41. y = sin x º º 2 2 π 42. y = cos 2x + º 2 2 π 43. y = 6 + 4 cos x º 2 1 44. y = tan x 2 π 45. y = 2 º tan x + 2 π 46. y = 2 tan x º 2 47. y = º1 + tan 2x 48. y = 3 + tan (x º π) π 1 49. y = 1 + tan 2x º 4 2 STUDENT HELP 1 32. y = 2 + sin x 2 WRITING EQUATIONS In Exercises 50–55, write an equation of the graph described. 50. The graph of y = sin 2πx translated down 5 units and right 2 units 51. The graph of y = 3 cos x translated up 3 units and left π units π 52. The graph of y = 5 tan 4x translated left unit and then reflected in the x-axis 4 1 53. The graph of y = sin 6x translated down 1 unit and then reflected in the line 3 y = º1 1 3 54. The graph of y = cos πx translated down units and left 1 unit, and then 2 2 3 reflected in the line y = º 2 π 1 55. The graph of y = 4 tan x translated up 6 units and right unit, and then 2 2 reflected in the line y = 6 56. CRITICAL THINKING Explain how the graph of y = sin x can be translated to become the graph of y = cos x. 57. HEIGHT OF A SWING A swing’s height h (in feet) above the ground is h = º8 cos † + 10 where the pivot is 10 feet above the ground, the rope is 8 feet long, and † is the angle that the rope makes with the vertical. Graph the function. What is the height of the swing when † is 45°? 8 ft 10 – h 8 ft † 10 ft h 58. BLOOD PRESSURE The pressure P (in millimeters of mercury) against the walls of the blood vessels of a certain person is given by 8π 3 P = 100 º 20 cos t where t is the time (in seconds). Graph the function. If one cycle is equivalent to one heartbeat, what is the person’s pulse rate in heartbeats per minute? 14.2 Translations and Reflections of Trigonometric Graphs 845 Page 7 of 8 ANIMAL POPULATIONS In Exercises 59 and 60, use the following information. Biologists use sine and cosine functions to model oscillations in predator and prey populations. The population R of rabbits and the population C of coyotes in a particular region can be modeled by Coyote population decreases. Rabbit population increases. π R = 25,000 + 15,000 cos t 12 π C = 5000 + 2000 sin t 12 where t is the time in months. Rabbit population decreases. Coyote population increases. 59. Graph both functions in the same coordinate plane and describe how the characteristics of the graphs relate to the diagram shown. 60. LOGICAL REASONING Look at the diagram above. Explain why each step leads to the next. 61. 62. AMUSEMENT PARK At an amusement park you watch your friend on a ride that simulates a free fall. You are standing 250 feet from the base of the ride and the ride is 100 feet tall. Write an equation that gives the distance d (in feet) that your friend has fallen as a function of the angle of elevation †. State the domain of the function. Then graph the function. WINDOW WASHERS You are standing 80 feet from a 300 foot building, watching as a window washer lowers himself to the ground. Write an equation that gives the window washer’s distance d (in feet) from the top of the building as a function of the angle of elevation †. State the domain of the function. Then graph the function. Not drawn to scale The Tower d your friend 100 d you † 250 ft Not drawn to scale window d washer 300 d you † 80 ft Test Preparation 63. MULTI-STEP PROBLEM You are at the top of a 120 foot building that straddles a road. You are looking down at a car traveling straight toward the building. a. Write an equation of the car’s distance from the base of the building as a function of the angle of depression from you to the car. b. Suppose the car is between you and a large road sign that you know is one mile (5280 feet) from the building. Write an equation for the distance between the road sign and the car as a function of the angle of depression from you to the car. c. Graph the functions you wrote in parts (a) and (b) in the same coordinate plane. d. Writing Describe how the graphs you drew in part (c) are geometrically related. ★ Challenge 846 64. FERRIS WHEEL Suppose a Ferris wheel has a radius of 20 feet and operates at a speed of 3 revolutions per minute. The bottom car is 4 feet above the ground. Write a model for the height of a person above the ground whose height when t = 0 is h = 44. Chapter 14 Trigonometric Graphs, Identities, and Equations Page 8 of 8 MIXED REVIEW CLASSIFYING CONICS Classify the conic section and write its equation in standard form. (Review 10.6 for 14.3) 65. 36x2 + 25y2 º 900 = 0 66. 9x2 º 16y2 º 144 = 0 67. 10x2 + 10y2 º 250 = 0 68. 100x2 + 81y2 º 100 = 0 COMBINATIONS Find the number of combinations. (Review 12.2) 69. 8C7 70. 142C1 71. 10C3 72. 14C5 73. 5C3 74. 6C2 75. 7C6 76. 100C2 EVALUATING FUNCTIONS Evaluate the six trigonometric functions of the angle †. (Review 13.1 for 14.3) 77. 78. 12 79. 10 26 † † † 10 3 20 80. † 5 81. 8 † 40 82. 5 9 16 † = 83. BOX LUNCHES You have made eight different lunches for eight people. How many different ways can you distribute the lunches? (Review 12.1) QUIZ 1 Self-Test for Lessons 14.1 and 14.2 Find the amplitude and period of the function. (Lesson 14.1) 5 1. y = sin 7x 2 2. y = cos 2x π 3. y = sin x 2 1 4. y = sin 2πx 4 5. y = 3 cos πx 3π 6. y = 4 cos x 2 7 7. y = cos 4x 3 1 8. y = sin x 3 1 9. y = 6 sin x 8 Graph the function. (Lessons 14.1, 14.2) 10. y = 2 sin πx 3 1 11. y = cos πx 2 2 12. y = ºsin 2x 1 13. y = 4 tan x 2 14. y = º4 + 2 sin 3x 15. y = 3 tan 2(x + π) 16. y = º3 cos (x + π) 1 π 17. y = º2 + cos (x º π) 18. y = 2 º 5 tan x + 2 3 19. GLASS ELEVATOR You are standing 120 feet from the base of a 260 foot building. You are looking at your friend who is going up the side of the building in a glass elevator. Write and graph a function that gives your friend’s distance d (in feet) above the ground as a function of her angle of elevation †. What is her angle of elevation when she is 70 feet above the ground? (Lesson 14.2) 14.2 Translations and Reflections of Trigonometric Graphs 847 Page 1 of 7 14.3 What you should learn GOAL 1 Use trigonometric identities to simplify trigonometric expressions and to verify other identities. Use trigonometric identities to solve real-life problems, such as comparing the speeds at which people pedal exercise machines in Example 7. GOAL 2 Why you should learn it RE FE To simplify real-life trigonometric expressions, such as the parametric equations that describe a carousel’s motion in Ex. 65. AL LI Verifying Trigonometric Identities GOAL 1 USING TRIGONOMETRIC IDENTITIES In this lesson you will use trigonometric identities to evaluate trigonometric functions, simplify trigonometric expressions, and verify other identities. ACTIVITY Developing Concepts Investigating Trigonometric Identities Use a graphing calculator to graph each side of the equation in the same viewing window. What do you notice about the graphs? Is the equation true for (a) no x-values, (b) some x-values, or (c) all x-values? (Set your calculator in radian mode and use º2π ≤ x ≤ 2π and º2 ≤ y ≤ 2.) 1. sin2 x + cos2 x = 1 2. sin (ºx) = ºsin x 3. sin x = ºcos x 4. cos x = 1.5 In the activity you may have discovered that some trigonometric equations are true for all values of x (in their domain). Such equations are called trigonometric identities. In Lesson 13.1 you used reciprocal identities to find the values of the cosecant, secant, and cotangent functions. These and other fundamental identities are listed below. F U N DA M E N TA L T R I G O N O M E T R I C I D E N T I T I E S RECIPROCAL IDENTITIES 1 sin † csc † = 1 cos † sec † = 1 tan † cot † = TANGENT AND COTANGENT IDENTITIES sin † cos † tan † = c os † sin † cot † = PYTHAGOREAN IDENTITIES sin2 † + cos2 † = 1 1 + tan2 † = sec2 † 1 + cot2 † = csc2 † COFUNCTION IDENTITIES π2 sin º † = cos † π2 cos º † = sin † π2 tan º † = cot † NEGATIVE ANGLE IDENTITIES sin (º†) = ºsin † 848 cos (º†) = cos † Chapter 14 Trigonometric Graphs, Identities, and Equations tan (º†) = ºtan † Page 2 of 7 Finding Trigonometric Values EXAMPLE 1 3 π Given that sin † = and < † < π, find the values of the other five trigonometric 5 2 functions of †. SOLUTION Begin by finding cos †. sin2 † + cos2 † = 1 Write Pythagorean identity. 35 + cos † = 1 2 3 5 2 Substitute }} for sin †. 35 cos2 † = 1 º 53 from each side. 2 Subtract }} 16 25 cos2 † = 2 Simplify. 4 5 Take square roots of each side. 4 5 Because † is in Quadrant II, cos † is negative. cos † = ± cos † = º Now, knowing sin † and cos †, you can find the values of the other four trigonometric functions. 3 4 º 5 cos † 4 cot † = = 3 = º3 sin † 5 3 sin † tan † = = = º 4 4 co s † º 5 5 1 sin † 11 1 5 3 1 cos † csc † = = = 3 EXAMPLE 2 5 1 5 4 sec † = = = º 4 º 5 Simplifying a Trigonometric Expression Simplify the expression sec † tan2 † + sec †. SOLUTION sec † tan2 † + sec † = sec † (sec2 † º 1) + sec † 3 EXAMPLE 3 Pythagorean identity = sec † º sec † + sec † Distributive property = sec3 † Simplify. Simplifying a Trigonometric Expression π2 Simplify the expression cos º x cot x. SOLUTION π cos º x cot x = sin x cot x 2 co s x = sin x sin x = cos x Cofunction identity Cotangent identity Simplify. 14.3 Verifying Trigonometric Identities 849 Page 3 of 7 You can use the fundamental identities on page 848 to verify new trigonometric identities. A verification of an identity is a chain of equivalent expressions showing that one side of the identity is equal to the other side. When verifying an identity, begin with the expression from one side and manipulate it algebraically until it is identical to the other side. Verifying a Trigonometric Identity EXAMPLE 4 Verify the identity cot (º†) = ºcot †. STUDENT HELP Study Tip Verifying an identity is not the same as solving an equation. When verifying an identity you should not use any properties of equality, such as adding the same number or expression to both sides. SOLUTION cos (º†) sin (º†) Cotangent identity = cos † ºsin † Negative angle identities = ºcot † Cotangent identity cot (º†) = Verifying a Trigonometric Identity EXAMPLE 5 cot2 x csc x Verify the identity = csc x º sin x. SOLUTION cot2 x csc2 x º 1 = csc x csc x c s c2 x csc x Pythagorean identity 1 csc x Write as separate fractions. = csc x º 1 csc x Simplify. = csc x º sin x Reciprocal identity = º EXAMPLE 6 Verifying a Trigonometric Identity sin x 1 º cos x 1 + cos x sin x Verify the identity = . SOLUTION STUDENT HELP Study Tip In Example 6, notice how multiplying by an expression equal to 1 allows you to write an expression in an equivalent form. 850 sin x (1 + cos x) sin x = 1 º cos x (1 º cos x)(1 + cos x) 1 + cos x Multiply by }} . 1 + cos x sin x (1 + cos x) 1 º cos x Simplify denominator. sin x (1 + cos x) sin x Pythagorean identity 1 + cos x sin x Simplify. = 2 = 2 = Chapter 14 Trigonometric Graphs, Identities, and Equations Page 4 of 7 GOAL 2 USING TRIGONOMETRIC IDENTITIES IN REAL LIFE In Lesson 13.7 you learned that parametric equations can be used to describe linear and parabolic motion. They can be used to describe other types of motion as well. Using Parametric Equations in Real Life EXAMPLE 7 INT STUDENT HELP NE ER T HOMEWORK HELP Visit our Web site www.mcdougallittell.com for extra examples. PHYSICAL FITNESS You and Sara are riding exercise machines that involve pedaling. The following parametric equations describe the motion of your feet and Sara’s feet: YOU: SARA: x = 8 cos 4πt x = 10 cos 2πt y = 8 sin 4πt y = 6 sin 2πt In each case, x and y are measured in inches and t is measured in seconds. a. Describe the paths followed by your feet and Sara’s feet. b. Who is pedaling faster (in revolutions per second)? SOLUTION a. Use the Pythagorean identity sin2 † + cos2 † = 1 to eliminate the parameter t. YOU: FOCUS ON APPLICATIONS SARA: x = cos 4πt 8 x = cos 2πt 10 y = sin 4πt 8 y = sin 2πt 6 cos2 4πt + sin2 4πt = 1 x8 + 8y = 1 2 Isolate the sine. cos2 2πt + sin2 2πt = 1 1x0 + 6y = 1 2 2 Pythagorean identity 2 y2 x2 + = 1 1 00 36 x2 + y2 = 64 Isolate the cosine. Substitute. Simplify. Your feet follow a circle with a radius of 8 inches. Sara’s feet follow an ellipse whose major axis is 20 inches long and whose minor axis is 12 inches long. b. The number of revolutions per second for you and Sara is the reciprocal of the common period of the corresponding parametric functions. YOU: RE FE L AL I 111 111 ==2 ELLIPTICAL TRAINERS provide excellent aerobic exercise by combining both lower and upper body movements. The smooth elliptical motion produces less impact than experienced on a treadmill. 2π 4π 1 2 SARA: 11 1 1 = = 1 2π 1 2π In one second, your feet travel around 2 times and Sara’s feet travel around 1 time. So, you are pedaling faster. ✓CHECK To check your results, set a graphing calculator to parametric, radian, and simultaneous modes. Enter both sets of parametric equations with 0 ≤ t ≤ 1 and a t-step of 0.01. As the paths are graphed, you can see that your path is traced faster. 14.3 Verifying Trigonometric Identities 851 Page 5 of 7 GUIDED PRACTICE ✓ Concept Check ✓ Vocabulary Check 1. What is a trigonometric identity? 2. Is sec (º†) equal to sec † or ºsec †? How do you know? 3. Verify the identity 1 º sin2 x cot2 x = sin2 x. Is there more than one way to verify the identity? If so, tell which way you think is easier and why. 4. ERROR ANALYSIS Describe what is wrong with the simplification shown. cos x º cos x sin2 x = cos x º cos x (1 + cos2 x) = cos x º cos x º cos3 x = ºcos3 x Skill Check ✓ Find the values of the other five trigonometric functions of †. 3 π 5. cos † = º, < † < π 5 2 2 π 6. tan † = , 0 < † < 3 2 4 3π 7. sec † = , < † < 2π 3 2 1 3π 8. sin † = º, π < † < 2 2 Simplify the expression. (sec x + 1)(sec x º 1) 9. tan x π 10. sin º x sec x 2 π 11. cos2 º x + cos2 (ºx) 2 Verify the identity. 1 12. = ºcsc x sin (ºx) 15. 13. cot x tan (ºx) = º1 14. csc x tan x = sec x PHYSICAL FITNESS Look back at Example 7 on page 851. Suppose your friend Pete starts riding another machine that involves pedaling. The motion of 5π 5π his feet is described by the equations x = 10 cos t and y = 6 sin t. What 2 2 type of path are his feet following? PRACTICE AND APPLICATIONS STUDENT HELP Extra Practice to help you master skills is on p. 959. 852 FINDING VALUES Find the values of the other five trigonometric functions of †. π 1 16. cos † = , 0 < † < 2 5 5 π 18. sin † = , 0 < † < 6 2 3 π 17. tan † = , 0 < † < 8 2 9 3π 20. cot † = º, < † < 2π 4 2 11 π 21. cos † = º, < † < π 12 2 7 π 22. csc † = , 0 < † < 5 2 10 3π 23. sec † = º, π < † < 3 2 1 π 24. tan † = º, < † < π 6 2 3π 25. sec † = 2, < † < 2π 2 5 3π 26. csc † = º, π < † < 3 2 3π 27. cot † = º3 , < † < 2π 2 3 π 19. sin † = , < † < π 5 2 Chapter 14 Trigonometric Graphs, Identities, and Equations Page 6 of 7 SIMPLIFYING EXPRESSIONS Simplify the expression. STUDENT HELP HOMEWORK HELP Example 1: Exs. 16–27 Examples 2, 3: Exs. 28–43 Examples 4–6: Exs. 44–53 Example 7: Exs. 61–64 28. cot x sec x cos (ºx) 29. sin (ºx) 31. sin x (1 + cot2 x) π 32. 1 º sin2 º x 2 tan º x 2 33. csc x sin (ºx) 35. + cos2 (ºx) csc x cos2 x tan2 (ºx) º 1 36. cos2 x π 34. cos º x csc x 2 2 π π π cos º x º 1 2 39. 1 + sin (ºx) tan º x sec x 2 38. 1 º csc2 x 2 37. sec x º tan x π sec x sin x + cos º x 2 41. 1 + sec x cot x cos x 40. π tan (ºx) sin 2 º x 30. sec x cos (ºx) º sin2 x π 43. tan º x cot x º csc2 x 2 42. cot2 x + sin2 x + cos2 (ºx) VERIFYING IDENTITIES Verify the identity. 44. cos x sec x = 1 45. tan x csc x cos x = 1 π 46. cos º x cot x = cos x 2 47. 2 º sec2 x = 1 º tan2 x 48. sin x + cos x cot x = csc x cos2 x + sin2 x 49. = cos2 x 1 + tan2 x sin2 (ºx) 50. = cos2 x tan2 x sin º x º 1 2 51. = º1 1 º cos (ºx) 1 + sin x cos x 52. + = 2 sec x cos x 1 + sin x cos (ºx) 53. = sec x + tan x 1 + sin (ºx) π IDENTIFYING CONICS Use a graphing calculator set in parametric mode to graph the parametric equations. Use a trigonometric identity to determine whether the graph is a circle, an ellipse, or a hyperbola. (Use a square viewing window.) 54. x = 6 cos t, y = 6 sin t 55. x = 5 sec t, y = tan t 56. x = 2 cos t, y = 3 sin t 57. x = 8 cos πt, y = 8 sin πt 58. x = 2 cot 2t, y = 3 csc 2t t t 59. x = cos , y = 4 sin 2 2 60. CRITICAL THINKING A function ƒ is odd if ƒ(ºx) = ºƒ(x). A function ƒ is even if ƒ(ºx) = ƒ(x). Which of the six trigonometric functions are odd? Which of them are even? 61. SHADOW OF A SUNDIAL The length s of a shadow cast by a vertical gnomon (column or shaft on a sundial) of height h when the angle of the sun above the horizon is † can be modeled by this equation: h sin (90° º †) sin † s = This equation was developed by Abu Abdullah al-Battani (circa A.D. 920). Show that the equation is equivalent to s = h cot †. Source: Trigonometric Delights 14.3 Verifying Trigonometric Identities 853 Page 7 of 7 GLEASTON WATER MILL In Exercises 62–64, use the following information. Suppose you have constructed a working scale model of the water wheel at the Gleaston Water Mill. The parametric equations that describe the motion of one of the paddles on each of the water wheels are as follows. Actual waterwheel: Scale model: x = 9 cos 8πt x = 0.5 cos 15πt y = 9 sin 8πt y = 0.5 sin 15πt In each case, x and y are measured in feet and t is measured in minutes. 62. Use a graphing calculator to graph both sets of parametric equations. 63. How is your scale model different from the actual waterwheel? 64. How many revolutions does each wheel make in 5 minutes? Test Preparation 65. MULTI-STEP PROBLEM The Dentzel Carousel in Glen Echo Park near Washington, D.C., is one of about 135 functioning antique carousels in the United States. The platform of the carousel is about 48 feet in diameter and makes about 5 revolutions per minute. Source: National Park Service a. Find parametric equations that describe the ride’s motion. b. Suppose the platform of the carousel had a diameter of 38 feet and made about 4.5 revolutions per minute. Find parametric equations that would describe the ride’s motion. 2 c. Writing How are the parametric equations affected when the speed is changed? How are the equations affected when the diameter is changed? ★ Challenge 66. Use the definitions of sine and cosine from Lesson 13.3 to derive the Pythagorean identity sin2 † + cos2 † = 1. 67. Use the Pythagorean identity sin2 † + cos2 † = 1 to derive the other Pythagorean identities, 1 + tan2 † = sec2 † and 1 + cot2 † = csc2 †. MIXED REVIEW QUADRATIC EQUATIONS Solve the equation by factoring. (Review 5.2 for 14.4) 68. x2 º 5x º 14 = 0 69. x2 + 5x º 36 = 0 70. x2 º 19x + 88 = 0 71. 2x2 º 7x º 15 = 0 72. 36x2 º 16 = 0 73. 9x2 º 1 = 0 EVALUATING EXPRESSIONS Evaluate the expression without using a calculator. Give your answer in both radians and degrees. (Review 13.4 for 14.4) º12 2 74. cosº1 2 75. tanº1 3 3 76. sinº1 º 2 1 77. sinº1 2 78. tanº1 (º1) 79. cosº1 GRAPHING Draw one cycle of the function’s graph. (Review 14.1) 854 80. y = 4 sin x 81. y = 2 cos x 82. y = tan 4πx 83. y = 3 tan πx 84. y = 5 cos 2x 85. y = 10 sin 4x Chapter 14 Trigonometric Graphs, Identities, and Equations Page 1 of 7 14.4 What you should learn GOAL 1 Solve a trigonometric equation. Solving Trigonometric Equations GOAL 1 SOLVING A TRIGONOMETRIC EQUATION In Lesson 14.3 you verified trigonometric identities. In this lesson you will solve trigonometric equations. To see the difference, consider the following equations: GOAL 2 Solve real-life trigonometric equations, such as an equation for the number of hours of daylight in Prescott, Arizona, in Example 6. sin2 x + cos2 x = 1 Equation 1 sin x = 1 Equation 2 Equation 1 is an identity because it is true for all real values of x. Equation 2, however, is true only for some values of x. When you find these values, you are solving the equation. Why you should learn it RE FE To solve many types of real-life problems, such as finding the position of the sun at sunrise in Ex. 58. AL LI Solving a Trigonometric Equation EXAMPLE 1 Solve 2 sin x º 1 = 0. SOLUTION First isolate sin x on one side of the equation. 2 sin x º 1 = 0 Write original equation. 2 sin x = 1 Add 1 to each side. 1 2 sin x = Divide each side by 2. 1 2 1 2 π 6 One solution of sin x = in the interval 0 ≤ x < 2π is x = sin–1 = . Another π 6 5π 6 such solution is x = π º = . Moreover, because y = sin x is a periodic function, there are infinitely many other solutions. You can write the general solution as π 6 x = + 2nπ or 5π 6 x = + 2nπ where n is any integer. ✓CHECK 1 You can check your answer graphically. Graph y = sin x and y = in the 2 same coordinate plane and find the points where the graphs intersect. You can see that there are infinitely many such points. y y 12 1 STUDENT HELP x Look Back For help with inverse trigonometric functions, see p. 792. π 2 y sin x 14.4 Solving Trigonometric Equations 855 Page 2 of 7 Solving a Trigonometric Equation in an Interval EXAMPLE 2 Solve 4 tan2 x º 1 = 0 in the interval 0 ≤ x < 2π. SOLUTION 4 tan2 x º 1 = 0 Write original equation. 4 tan2 x = 1 Add 1 to each side. 1 4 tan2 x = Divide each side by 4. 1 2 tan x = ± Take square roots of each side. Use a calculator to find values of x for tan-1 (.5) .463647609 tan-1 (-.5) -.463647609 1 2 which tan x = ±, as shown at the right. The general solution of the equation is STUDENT HELP Study Tip Note that to find the general solution of a trigonometric equation, you must add multiples of the period to the solutions in one cycle. x ≈ 0.464 + nπ or x ≈ º0.464 + nπ where n is any integer. The solutions that are in the interval 0 ≤ x < 2π are: x ≈ 0.464 x ≈ 0.464 + π ≈ 3.61 x ≈ º0.464 + π ≈ 2.628 x ≈ º0.464 + 2π ≈ 5.82 ✓CHECK Check these solutions by substituting them back into the original equation. EXAMPLE 3 Factoring to Solve a Trigonometric Equation Solve sin2 x cos x = 4 cos x. SOLUTION sin2 x cos x = 4 cos x sin2 x cos x º 4 cos x = 0 Write original equation. Subtract 4 cos x from each side. 2 cos x (sin x º 4) = 0 Factor out cos x. cos x (sin x + 2)(sin x º 2) = 0 Factor difference of squares. Set each factor equal to 0 and solve for x, if possible. cos x = 0 STUDENT HELP Study Tip Remember not to divide both sides of an equation by a variable expression, such as cos x. 856 sin x + 2 = 0 3π π x = or x = 2 2 sin x º 2 = 0 sin x = º2 sin x = 2 Because neither sin x = º2 nor sin x = 2 has a solution, the only solutions in the 3π 2 π 2 interval 0 ≤ x < 2π are x = and x = . π 2 3π 2 The general solution is x = + 2nπ or x = + 2nπ where n is any integer. Chapter 14 Trigonometric Graphs, Identities, and Equations Page 3 of 7 Using the Quadratic Formula EXAMPLE 4 Solve cos2 x º 4 cos x + 1 = 0 in the interval 0 ≤ x ≤ π. SOLUTION STUDENT HELP Look Back For help with the quadratic formula, see p. 291. Since the equation is in the form au2 + bu + c = 0, you can use the quadratic formula to solve for u = cos x. cos2 x º 4 cos x + 1 = 0 Write original equation. 4 ± (º 4 )2 º4(1 )( 1) 2(1) Quadratic formula = = 2 ± 3 4 ± 12 2 Simplify. ≈ 3.73 or 0.268 Use a calculator. cos x = x ≈ cosº1 3.73 or x ≈ cosº1 0.268 Use inverse cosine. No solution Use a calculator if possible. ≈ 1.30 In the interval 0 ≤ x ≤ π, the only solution is x ≈ 1.30. Check this in the original equation. .......... When solving a trigonometric equation, it is possible to obtain extraneous solutions. Therefore, you should always check your solutions in the original equation. An Equation with Extraneous Solutions EXAMPLE 5 INT STUDENT HELP NE ER T HOMEWORK HELP Visit our Web site www.mcdougallittell.com for extra examples. Solve 1 º cos x = 3 sin x in the interval 0 ≤ x < 2π. SOLUTION 1 º cos x = 3 sin x Write original equation. (1 º cos x) 2 = (3 sin x)2 1 º 2 cos x + cos2 x = 3 sin2 x Multiply. 1 º 2 cos x + cos2 x = 3(1 º cos2 x) Pythagorean identity 4 cos2 x º 2 cos x º 2 = 0 Quadratic form 2 cos2 x º cos x º 1 = 0 Divide each side by 2. (2 cos x + 1)(cos x º 1) = 0 Factor. 2 cos x + 1 = 0 or cos x º 1 = 0 2π 3 cos x = º 1 2 cos x = 1 4π 3 x=0 x = or x = Square both sides. 4π 3 Zero product property Solve for cos x. Solve for x. The apparent solution x = does not check in the original equation. The only 2π 3 solutions in the interval 0 ≤ x < 2π are x = 0 and x = . 14.4 Solving Trigonometric Equations 857 Page 4 of 7 FOCUS ON CAREERS GOAL 2 SOLVING TRIGONOMETRIC EQUATIONS IN REAL LIFE Solving a Real-Life Trigonometric Equation EXAMPLE 6 METEOROLOGY The number h of hours of sunlight per day in Prescott, Arizona, can be modeled by π 6 h = 2.325 sin (t º 2.667) + 12.155 RE FE L AL I METEOROLOGIST INT Operational meteorologists, the largest group of specialists in the field, use complex computer models to forecast the weather. Other types of meteorologists include physical meteorologists and climatologists. where t is measured in months and t = 0 represents January 1. On which days of the year are there 13 hours of sunlight in Prescott? Source: Gale Research Company SOLUTION Method 1 Substitute 13 for h in the model and solve for t. π 6 2.325 sin (t º 2.667) + 12.155 = 13 π 6 2.325 sin (t º 2.667) = 0.845 NE ER T π 6 CAREER LINK sin (t º 2.667) ≈ 0.363 www.mcdougallittell.com π π (t º 2.667) ≈ sinº1 (0.363) or (t º 2.667) ≈ π º sinº1 (0.363) 6 6 π (t º 2.667) ≈ 0.371 6 π (t º 2.667) ≈ π º 0.371 ≈ 2.771 6 t º 2.667 ≈ 0.709 t º 2.667 ≈ 5.292 t ≈ 3.38 t ≈ 7.96 The time t = 3.38 represents 3 full months plus (0.38)(30) ≈ 11 days, or April 11. Likewise, the time t = 7.96 represents 7 full months plus (0.96)(31) ≈ 30 days, or August 30. (Notice that these two days occur about 70 days before and after June 21, which is the date of the summer solstice, the longest day of the year.) Method 2 Use a graphing calculator. Graph the equations π 6 y = 2.325 sin (x º 2.667) + 12.155 y = 13 in the same viewing window. Then use the Intersect feature to find the points of intersection. Intersection X=3.3773887 Y=13 858 Intersection X=7.9566113 Y=13 From the screens above, you can see that t ≈ 3.38 or t ≈ 7.96. The time t = 3.38 is about April 11, and the time t = 7.96 is about August 30. Chapter 14 Trigonometric Graphs, Identities, and Equations Page 5 of 7 GUIDED PRACTICE Vocabulary Check Concept Check ✓ ✓ 1. What is the difference between a trigonometric equation and a trigonometric identity? 2. Name several techniques for solving trigonometric equations. 3. ERROR ANALYSIS Describe the error(s) in the calculations shown. 1 2 1 cos x = 2 cos2 x = cos x π 3 5π 3 Solution for 0 ≤ x < 2π 5π 3 General solution x = or x = π 3 x = + 2π or x = + 2π Skill Check ✓ Solve the equation in the interval 0 ≤ x < 2π. 4. 2 cos x + 4 = 5 5. 3 sec2 x º 4 = 0 6. tan2 x = cos x tan2 x 7. 5 cos x º 3 = 3 cos x Find the general solution of the equation. 9. 4 sin x = 3 8. 3 csc x + 5 = 0 2 2 10. 1 + tan x = 6 º 2 sec x 12. 11. 2 º 2 cos2 x = 5 sin x + 3 METEOROLOGY Look back at the model in Example 6 on page 858. On which days of the year are there 10 hours of sunlight in Prescott, Arizona? PRACTICE AND APPLICATIONS STUDENT HELP Extra Practice to help you master skills is on p. 960. CHECKING SOLUTIONS Verify that the given x-value is a solution of the equation. π 15. 4 cos2 x º 3 = 0, x = 6 5π 14. csc x º 2 = 0, x = 6 π 3 16. 3 tan x º 3 = 0, x = 4 5π 17. 2 sin4 x º sin2 x = 0, x = 4 13π 18. 2 cot4 x º cot2 x º 15 = 0, x = 6 13. 5 + 4 cos x º 1 = 0, x = π SOLVING Find the general solution of the equation. 19. 2 cos x º 1 = 0 20. 3 tan x º 3 =0 21. sin x = sin (ºx) + 1 22. 4 cos x = 2 cos x + 1 STUDENT HELP 23. 4 sin2 x º 2 = 0 24. 9 tan2 x º 3 = 0 HOMEWORK HELP 25. sin x cos x º 2 cos x = 0 26. 2 cos x sin x º cos x = 0 Examples 1, 3: Exs. 13–32 Examples 2, 4, 5: Exs. 13–18, 33–56 Example 6: Exs. 57–59 π 3 5π 3 2 27. 2 sin x º sin x = 1 28. 0 = cos2 x º 5 cos x + 1 29. 1 º sin x = 3 cos x 30. si n x = 2 sin x º 1 31. cos x º 1 = ºcos x 32. 6 sin x = sin x + 3 31. }} + 2nπ, }} + 2nπ 14.4 Solving Trigonometric Equations 859 Page 6 of 7 SOLVING Solve the equation in the interval 0 ≤ x < 2π. Check your solutions. 33. 5 cos x º 3 = 0 34. 3 sin x = sin x º 1 35. tan2 x º 3 = 0 36. 10 tan x º 5 = 0 2 37. 2 cos x º sin x º 1 = 0 38. cos3 x = cos x 39. sec2 x º 2 = 0 40. tan2 x = sin x sec x 41. 2 cos x = sec x 42. cos x csc2 x + 3 cos x = 7 cos x APPROXIMATING SOLUTIONS Use a graphing calculator to approximate the solutions of the equation in the interval 0 ≤ x < 2π. 43. 3 tan x + 1 = 13 44. 8 cos x + 3 = 4 45. 4 sin x = º2 sin x º 5 46. 3 sin x + 5 cos x = 4 FINDING INTERCEPTS Find the x-intercepts of the graph of the given function in the interval 0 ≤ x < 2π. 47. y = 2 sin x + 1 48. y = 2 tan2 x º 6 49. y = sec2 x º 1 50. y = º3 cos x + sin x FINDING INTERSECTION POINTS Find the points of intersection of the graphs of the given functions in the interval 0 ≤ x < 2π. tan2 x 51. y = 3 y = 3 º 2 tan x 54. y = sin2 x y = 2 sin x º 1 57. 52. y = 9 cos2 x 53. y = tan x sin x y = cos2 x + 8 cos x º 2 y = cos x 56. y = 4 cos2 x 55. y = 2 º sin x tan x y = cos x y = 4 cos x º 1 OCEAN TIDES The tide, or depth of the ocean near the shore, changes throughout the day. The depth of the Bay of Fundy can be modeled by π 6.2 d = 35 º 28 cos t where d is the water depth in feet and t is the time in hours. Consider a day in which t = 0 represents 12:00 A.M. For that day, when do the high and low tides FOCUS ON APPLICATIONS 1 2 occur? At what time(s) is the water depth 3 feet? 58. POSITION OF THE SUN Cheyenne, Wyoming, has a latitude of 41°N. At this latitude, the position of the sun at sunrise can be modeled by 326π5 D = 31 sin t º 1.4 where t is the time in days and t = 1 represents January 1. In this model, D represents the number of degrees north or south of due east that the sun rises. Use a graphing calculator to determine the days that the sun is more than 20° north of due east at sunrise. RE FE L AL I THE BAY OF FUNDY is the portion of the Atlantic Ocean that runs along the southern coast of New Brunswick in Canada. It is the site of the highest tides in the world. INT NE ER T APPLICATION LINK www.mcdougallittell.com 860 59. METEOROLOGY A model for the average daily temperature T (in degrees Fahrenheit) in Kansas City, Missouri, is given by 21π2 T = 54 + 25.2 sin t + 4.3 where t is measured in months and t = 0 represents January 1. What months have average daily temperatures higher than 70°F? Do any months have average daily temperatures below 20°F? Source: National Climatic Data Center Chapter 14 Trigonometric Graphs, Identities, and Equations Page 7 of 7 Test Preparation 60. MULTIPLE CHOICE What is the general solution of the equation cos x + 2 = ºcos x? Assume n is an integer. A ¡ x = + 2nπ π 4 B ¡ x = + 2nπ or x = + 2nπ C ¡ x = + 2nπ 3π 4 D ¡ x = + 2nπ or x = + 2nπ E ¡ x = + nπ or x = + nπ 3π 4 π 4 7π 4 3π 4 5π 4 5π 4 61. MULTIPLE CHOICE Find the points of intersection of the graphs of y = 2 + sin x and y = 3 º sin x in the interval 0 ≤ x < 2π. π6, 52, 56π, 52 ¡ π3, 12, 43π, 12 ★ Challenge π6, 12, 56π, 12 11π 5 , ¡ π6, 52, 6 2 A ¡ B ¡ D E C ¡ π3, 52, 43π, 52 MATRICES In Exercises 62 and 63, use the following information. Matrix multiplication can be used to rotate a point (x, y) counter clockwise about the origin through an angle †. The coordinates of the resulting point (x§, y§) are determined by the following matrix equation: cos † ºsin † sin † cos † x x§ = y y§ y 62. The point (4, 1) is rotated counter clockwise about π the origin through an angle of . What are the 3 † (x ´, y ´) coordinates of the resulting point? EXTRA CHALLENGE (x, y) 63. Through what angle † must the point (2, 4) be x rotated to produce (x§, y§) = (º2 + 3, 1 + 23 )? www.mcdougallittell.com MIXED REVIEW USING COMPLEMENTS Two six-sided dice are rolled. Find the probability of the given event. (Review 12.4) 64. The sum is less than or equal to 10. 65. The sum is not 4. 66. The sum is not 2 or 12. 67. The sum is greater than 3. GRAPHING Graph the function. (Review 14.1, 14.2 for 14.5) 68. y = sin 3x 69. y = 2 cos 4x 70. y = 10 sin 2x 1 71. y = 3 cos x 2 1 72. y = tan x 4 1 73. y = 3 tan x º 2 2 1 74. y = cos x + π 2 3π 75. y = 3 + tan x º 2 77. 76. y = ºsin 3π(x + 4) + 1 43 ft SURVEYING Suppose you are trying to determine the width w of a small pond. You stand at a point 43 feet from one end of the pond and 50 feet from the other end. The angle formed by your lines of sight to each end of the pond measures 45°. How wide is the pond? (Review 13.6) 45 you w 50 ft 14.4 Solving Trigonometric Equations 861 Page 1 of 7 E X P L O R I N G DATA A N D S TAT I S T I C S 14.5 What you should learn Model data with a sine or cosine function. Modeling with Trigonometric Functions GOAL 1 Graphs of sine and cosine functions are called sinusoids. When you write a sine or cosine function for a sinusoid, you need to find the values of a, b > 0, h, and k for y = a sin b(x º h) + k GOAL 1 GOAL 2 Use technology to write a trigonometric model, as applied in Example 4. Why you should learn it RE or y = a cos b(x º h) + k 2π where |a| is the amplitude, is the period, h is the horizontal shift, and k is the b vertical shift. EXAMPLE 1 Writing Trigonometric Functions Write a function for the sinusoid. π3 , 4 y a. b. (0.5, 5) 5 2 x x 0.5 π 3 FE To model many types of real-life quantities, such as the temperature inside and outside an igloo in Ex. 30. AL LI WRITING A TRIGONOMETRIC MODEL (1.5, 15) (0, 4) SOLUTION a. Since the maximum and minimum values of the function occur at points equidistant from the x-axis, the curve has no vertical shift. And because the minimum occurs on the y-axis, the graph is a reflection of a cosine curve with no horizontal shift. Therefore, the function has the form y = a cos bx. 2π b 2π 3 The period is = , so b = 3. The amplitude is |a| = 4. Since the graph is a reflection, a = º4. The function is y = º4 cos 3x. b. Since the maximum and minimum values of the function do not occur at points equidistant from the x-axis, the curve has a vertical shift. To find the value of k, add the maximum and minimum values and divide by 2: M+m 2 5 + (º15) 2 º10 2 k = = = = º5 Because the graph crosses the y-axis at y = k, the graph is a sine curve with no horizontal shift. Therefore, the function has the form y = a sin bx º 5. 2π b The period is = 2, so b = π. INT STUDENT HELP NE ER T HOMEWORK HELP Visit our Web site www.mcdougallittell.com for extra examples. 862 Mºm 2 5 º (º15) 2 20 2 The amplitude is |a| = = = = 10. Since the graph is not a reflection, a = 10 > 0. The function is y = 10 sin πx º 5. Chapter 14 Trigonometric Graphs, Identities, and Equations Page 2 of 7 RE FE L AL I Meteorology EXAMPLE 2 Modeling a Sinusoid Write a trigonometric model for the average daily temperature in Birmingham, Alabama. Source: National Climatic Data Center Temperature (F) Daily Temperature in Birmingham T 80 60 40 20 0 0 2 t 6 8 10 4 Months since January 1 SOLUTION Notice that the graph crosses the T-axis at the minimum point. So if you model the temperature curve with a cosine function, there is a reflection but no horizontal shift. The mean of the maximum and minimum values is 60, so there is a vertical shift of k = 60. 2π b π 6 The period is = 12, so b = . The amplitude is |a| = 20, and because the graph is a reflection it follows that a = º20. RE FE L AL I Ferris Wheel π The model is T = º20 cos t + 60 where t is measured in months and t = 0 6 represents January 1. EXAMPLE 3 Modeling Circular Motion A Ferris wheel with a radius of 25 feet is rotating at a rate of 3 revolutions per minute. When t = 0, a chair starts at the lowest point on the wheel, which is 5 feet above the ground. Write a model for the height h (in feet) of the chair as a function of the time t (in seconds). SOLUTION When the chair is at the bottom of the Ferris wheel, it is 5 feet above the ground, so m = 5. When it is at the top, it is 5 + 2(25) = 55 feet above the ground, so M = 55. M+m 2 55 + 5 2 60 2 The vertical shift for the model is k = = = = 30. When t = 0, the height is at its minimum, so the model is a cosine function with a < 0 and no horizontal shift. Mºm 55 º 5 50 The amplitude is |a| = = = = 25. Because a < 0 it follows that 2 2 2 a = º25. Since the Ferris wheel is rotating at 3 revolutions per minute, it completes one 2π b π 10 revolution in 20 seconds. The period is = 20, so b = . π 10 A model for the height of the chair as a function of time is h = º25 cos t + 30. 14.5 Modeling with Trigonometric Functions 863 Page 3 of 7 GOAL 2 SINUSOIDAL MODELING USING TECHNOLOGY There are two ways you can model a set of data points whose scatter plot appears sinusoidal. One way is to estimate the minimum and maximum values and use the technique shown in Example 2. Another way is to use a graphing calculator that has a sinusoidal regression feature. The advantage of the second method is that it uses all of the data points to find the model. RE FE L AL I Meteorology EXAMPLE 4 Using Sinusoidal Regression The average daily temperature T (in degrees Fahrenheit) in Fairbanks, Alaska, is given in the table. Time t is measured in months, with t = 0 representing January 1. Write a trigonometric model that gives T as a function of t. 0.5 2.5 3.5 4.5 5.5 6.5 7.5 8.5 9.5 10.5 11.5 º10.1 º3.6 11.0 30.7 48.6 59.8 62.5 56.8 45.5 25.1 2.7 º6.5 t T 1.5 Source: National Climatic Data Center SOLUTION Begin by entering the data in a graphing calculator and drawing a scatter plot. L1 L2 .5 -10.1 1.5 -3.6 2.5 11 3.5 30.7 4.5 48.6 L1(1)=.5 1 L3 Enter the data. 2 Draw a scatter plot. Because the scatter plot appears sinusoidal, you can fit the data with a sine function to get the following model: T = 37.4 sin (0.518t º 1.72) + 26.5 SinReg y=a*sin(bx+c)+d a=37.40756055 b=.5184527958 c=-1.717020267 d=26.50468918 3 Perform a sinusoidal regression. 4 Graph the model and the data in the same viewing window. After obtaining the model, you should graph it in the same viewing window as the scatter plot to see how well the model fits the data. In this case, the model is a good fit. ✓CHECK As a check on the reasonableness of the model, notice that the period 2π is ≈ 12, which is the number of months in a year. 0.518 864 Chapter 14 Trigonometric Graphs, Identities, and Equations Page 4 of 7 GUIDED PRACTICE ✓ Concept Check ✓ Vocabulary Check ?. 1. Complete this statement: Graphs of sine and cosine functions are called 2. Which two points are most useful when writing a sinusoidal model for a given graph or set of data? Explain. 3. Describe a characteristic of a sinusoidal graph that you would model with a cosine function rather than a sine function. Skill Check ✓ Write a function for the sinusoid. y 4. y 5. π ,1 2 2 (0, 1) x π 3 π , 1 6 x 1 (1, 3) 4 Write a function for the sinusoid with maximum at A and minimum at B. 6. A(0, 8), B(π, º2) 9. 7. A(π, 10), B(3π, 4) 8. A(6, 5), B(2, 1) FERRIS WHEEL Look back at Example 3. Suppose the Ferris wheel rotates at a rate of 4 revolutions per minute and has a radius of 20 feet. Write a model for the height h (in feet) of the chair as a function of the time t (in seconds). PRACTICE AND APPLICATIONS STUDENT HELP WRITING FUNCTIONS Write a function for the sinusoid. Extra Practice to help you master skills is on p. 960. 10. y 3 π4 , 5 11. (0, 3) 5 x x π 2 π 4 3π4 , 5 π4 , 3 12. y y 13. π ,2 2 3 (1, 4) 4 x π 3 (0, 4) π6 , 2 STUDENT HELP 14. Example 1: Example 2: Example 3: Example 4: Exs. 10–27 Exs. 30, 32 Exs. 31, 33 Exs. 34, 35 2 14 , 2 3 ,0 4 y 15. y HOMEWORK HELP x 1 2 (0, 2) 3 x x 1 (2, 6) 14.5 Modeling with Trigonometric Functions 865 Page 5 of 7 WRITING TRIGONOMETRIC FUNCTIONS Write a function for the sinusoid with maximum at A and minimum at B. 16. A(0, 3), B(2π, º3) 17. A(π, 8), B(3π, º8) 18. A(9π, 1), B(3π, º5) π 19. A , 4 , B(0, 2) 3 20. A(2, 6), B(6, 0) 21. A(3, 7), B(1, º3) 2π 22. A, 9, B(2π, 5) 3 25. A(0, 0), B(3π, º8) π 23. A , 11 , B(0, º1) 6 24. A(0, 5), B(4, º13) 26. A(6, 2), B(0, º4) π π 27. A , º2 , B , º6 4 12 28. CRITICAL THINKING Any sinusoid can be modeled by either a sine function or a cosine function. Using the graph from part (a) of Example 1, find the values of a, b, h, and k for the model y = a sin b(x º h) + k. Use identities to show that the model you found is equivalent to the model from part (a) of Example 1. 29. Writing Since any sinusoid can be modeled by either a sine function or a cosine function, you can choose the type of function that is more convenient. For a sinusoid whose y-intercept occurs halfway between the maximum and minimum values of the function, tell which type of function you would use to model the graph. Explain your answer. CLIMATE CONTROL Eskimos use igloos as temporary shelter from harsh winter weather. The graph shows the temperatures inside and outside an igloo throughout a typical winter day. Write a sinusoidal model for the outside temperature T (in degrees Fahrenheit) as a function of the time of day t (in hours since midnight). Then write sinusoidal models for the floor-level temperature and for the sleepingplatform temperature. Igloo Temperatures T 40 Temperature (F) 30. 20 0 4 8 12 16 20 20 Hours since 8 A.M. Source: Scientific American FOCUS ON APPLICATIONS RE FE L AL I THE S.S. BEAVER was the first steamship on the Pacific Coast. It was built in the 1830s and sailed out of British Columbia, Canada, for over 50 years. 866 31. STEAMSHIPS The paddle wheel of the S.S. Beaver was 13 feet in diameter and revolved 30 times per minute when moving at top speed. Using this speed and starting from a point at the very top of the wheel, write a model for the height h (in feet) of the end of a paddle relative to the water’s surface as a function of the time t (in minutes). (Assume the paddle is 2 feet below the water’s surface at its lowest point.) Source: S.S. Beaver: The Ship That Saved the West 32. OCEAN TIDES The height of the water in a bay varies sinusoidally over time. On a certain day off the coast of Maine, a high tide of 10 feet occurred at 5:00 A.M. and a low tide of 2 feet occurred at 1:00 P.M. Write a model for the height h (in feet) of the water as a function of time t (in hours since midnight). 33. SEWING MACHINES In front of the Antique Sewing Machine Museum in Arlington, Texas, is the largest sewing machine in the world. The flywheel, which turns as the machine sews, is 5 feet in diameter. Write a model for the height h (in feet) of the handle on the flywheel as a function of the time t (in seconds), assuming that the wheel makes a complete revolution every 2 seconds and that the handle starts at its minimum height of 4 feet above the ground. Chapter 14 Trigonometric Graphs, Identities, and Equations t Page 6 of 7 In Exercises 34 and 35, use a graphing calculator to find a sinusoidal model that fits the data. STATISTICS CONNECTION METEOROLOGY The average daily temperature T (in degrees Fahrenheit) in 34. Moline, Illinois, is given in the table. Time t is measured in months, with t = 0 representing January 1. Find a model for the data. Source: National Climatic Data Center t 0.5 1.5 2.5 3.5 4.5 5.5 6.5 7.5 8.5 9.5 10.5 11.5 T 19.9 24.8 37.4 50.4 61.4 71.1 75.2 72.7 64.7 53.0 39.6 25.4 HEATING DEGREE-DAYS For any given day, the number of degrees that 35. the average temperature is below 65°F is called the degree-days for that day. This figure is used to calculate how much is spent on heating. The table below gives the total number T of degree-days for each month t in Dubuque, Iowa, with t = 1 representing January. Find a model for the data. Source: Workshop Math 1 t T Test Preparation 2 3 1420 1204 1026 4 5 6 7 8 9 10 11 12 546 260 78 12 31 156 450 906 1287 36. MULTIPLE CHOICE During one cycle, a sinusoid has a minimum at (18, 44) and a maximum at (30, 68). What is the amplitude of this sinusoid? A ¡ 6 B ¡ C ¡ 12 22 D ¡ 24 E ¡ 48 37. MULTIPLE CHOICE During one cycle, a sinusoid has a maximum at (4, 12) and a minimum at (12, º2). What is the period of this sinusoid? ★ Challenge A ¡ 8 B ¡ C ¡ 8π 16 D ¡ 16π E ¡ 32 38. FINDING A FUNCTION Write a sinusoidal function whose graph has a π2 minimum at (π, 4) and a maximum at , 7 . MIXED REVIEW FINDING PROBABILITY Two six-sided dice are rolled. Find the probability of the given event. (Review 12.4) 39. The sum is 3. 40. The sum is 6. 41. The sum is 12. 42. The sum is odd. 43. The sum is 5 or 6. 44. The sum is less than 7. EVALUATING FUNCTIONS Evaluate the function without using a calculator. (Review 13.3 for 14.6) 45. cos (º225°) 46. cos 240° 47. sin (º30°) 48. tan 330° 7π 49. tan º 6 5π 50. sin 4 FINDING AREA Find the area of ¤ABC having the given side lengths. (Review 13.6) 51. a = 3, b = 8, c = 10 52. a = 12, b = 20, c = 25 53. a = 5, b = 9, c = 11 54. a = 4, b = 6, c = 7 55. a = 6, b = 11, c = 14 56. a = 8, b = 13, c = 20 14.5 Modeling with Trigonometric Functions 867 Page 7 of 7 QUIZ 2 Self-Test for Lessons 14.3–14.5 Simplify the expression. (Lesson 14.3) π 1. tan º x sec x 2 co s x 2. 1 º sin2 x + sec x π 3. cos (ºx) cos º x 2 Find the general solution of the equation. (Lesson 14.4) 4. 4 cos2 x º 3 = 0 6. 3 tan2 x + 4 tan x = º3 5. 3 sin2 x º 8 sin x = 3 Write a function for the sinusoid with maximum at A and minimum at B. (Lesson 14.5) 3π π 7. A , 5 , B , º5 4 4 10. 8. A(0, 3), B(3π, 1) 9. A(π, 6), B(0, 2) METEOROLOGY The average daily temperature T (in degrees Fahrenheit) in Detroit, Michigan, is given in the table. Time t is measured in months, with t = 0 representing January 1. Find a model for the data. (Lesson 14.5) 0.5 1.5 2.5 3.5 4.5 5.5 6.5 7.5 8.5 9.5 10.5 11.5 T 22.9 25.4 35.7 47.3 58.4 67.6 72.3 70.5 63.2 51.2 40.2 28.3 INT t NE ER T Music and Math THEN NOW APPLICATION LINK www.mcdougallittell.com MUSIC AND MATH have been studied together for many centuries. For instance, 2500 years ago Pythagoras discovered that when the ratio of the lengths of two strings is a whole number, plucking the strings produces harmonious tones. TODAY we know that musical notes can be modeled by sine functions. The function y = sin 2πƒx models a note with frequency ƒ (in hertz) where x represents the length of time (in seconds) that the note is played. By adding the functions modeling two different notes, you can analyze the sound that occurs when the notes are played together. 1. Write functions that model notes with frequencies 10 hertz, 11 hertz, 12 hertz, 13 hertz, 14 hertz, and 15 hertz. Use a graphing calculator to graph the sum of the models for 10 hertz and 15 hertz. How many cycles of the function occur in 1 second? 2. 3. How many cycles per second are there in the function that describes what you hear when notes with frequencies of 10 and 14 hertz are played together? 10 and 13 hertz? 10 and 12 hertz? Harps first appeared. c. 3000 B . C . First piano built. 1964 c. A . D . 700 1709 The lyre is introduced. 868 Chapter 14 Trigonometric Graphs, Identities, and Equations Synthesizer invented. Page 1 of 6 14.6 What you should learn GOAL 1 Evaluate trigonometric functions of the sum or difference of two angles. Use sum and difference formulas to solve real-life problems, such as determining when pistons in a car engine are at the same height in Example 6. GOAL 2 Why you should learn it RE GOAL 1 SUM AND DIFFERENCE FORMULAS In this lesson you will study formulas that allow you to evaluate trigonometric functions of the sum or difference of two angles. SUM AND DIFFERENCE FORMULAS SUM FORMULAS DIFFERENCE FORMULAS sin (u + v) = sin u cos v + cos u sin v sin (u º v) = sin u cos v º cos u sin v cos (u + v) = cos u cos v º sin u sin v cos (u º v) = cos u cos v + sin u sin v tan u + tan v tan (u + v) = 1 º tan u tan v tan u º tan v tan (u º v) = 1 + tan u tan v In general, sin (u + v) ≠ sin u + sin v. Similar statements can be made for the other trigonometric functions of sums and differences. FE To model real-life quantities, such as the size of an object in an aerial photograph in Ex. 58. AL LI Using Sum and Difference Formulas Evaluating a Trigonometric Expression EXAMPLE 1 Find the exact value of (a) cos 75° and (b) tan π. 12 SOLUTION a. cos 75° = cos (45° + 30°) Substitute 45° + 30° for 75°. = cos 45° cos 30° º sin 45° sin 30° Sum formula for cosine 2 3 2 1 º = 2 2 2 2 Evaluate. 6 º 2 = 4 Simplify. π π π b. tan = tan º 3 4 12 π π 3 π 12 π tan º tan 3 4 = π π 1 + tan tan 3 3 º 1 Difference formula for tangent 4 = Evaluate. = 2 º 3 Simplify. 1 + (3 )(1) ✓CHECK π 4 Substitute }} º }} for }}. Try checking these results with a calculator. For instance, evaluate 6 º 2 cos 75° and to see that both have the same value. 4 14.6 Using Sum and Difference Formulas 869 Page 2 of 6 Using a Difference Formula EXAMPLE 2 3 3π 12 Find sin (u º v) given that sin u = º with π < u < and cos v = with 5 2 13 π 0 < v < . 2 SOLUTION 4 5 5 13 Using a Pythagorean identity and quadrant signs gives cos u = º and sin v = . sin (u º v) = sin u cos v º cos u sin v Difference formula for sine 45 153 3 12 5 13 Substitute. 16 65 Simplify. = º º º = º Simplifying an Expression EXAMPLE 3 Simplify the expression cos (x º π). SOLUTION cos (x º π) = cos x cos π + sin x sin π = (cos x)(º1) + (sin x)(0) Evaluate. = ºcos x Simplify. Solving a Trigonometric Equation EXAMPLE 4 Difference formula for cosine π 4 π4 Solve sin x + + 1 = sin º x for 0 ≤ x < 2π. SOLUTION π4 π 4 π 4 π 4 π 4 π 4 π 4 Write original equation. sin x + + 1 = sin º x π 4 sin x cos + cos x sin + 1 = sin cos x º cos sin x π 4 Use formulas. π 4 sin x cos + cos x sin + 1 = cos x sin º sin x cos Commutative property π 4 2 sin x cos = º1 Simplify. 2 Evaluate. 2(sin x) 2 = º1 1 2 2 sin x = º = º 2 5π 4 Solve for sin x. 7π 4 In the interval 0 ≤ x < 2π, the solutions are x = and x = . ✓CHECK You can check the solutions with a graphing calculator by graphing each side of the original equation and using the Intersect feature to determine the x-values for which the expressions are equal. 870 Chapter 14 Trigonometric Graphs, Identities, and Equations Page 3 of 6 GOAL 2 RE FE L AL I Biomechanics SUM AND DIFFERENCE FORMULAS IN REAL LIFE EXAMPLE 5 Simplifying a Real-Life Formula The force F (in pounds) on a person’s back when he or she bends over at an angle † is 0.6W sin († + 90°) sin 12° F = where W is the person’s weight (in pounds). Simplify this formula. † SOLUTION Begin by expanding sin († + 90°). 0.6W sin († + 90°) sin 12° 0.6W(sin † cos 90° + cos † sin 90°) 0.208 F = ≈ 0.6 0.208 = W [(sin †)(0) + (cos†)(1)] ≈ 2.88W cos † EXAMPLE 6 Solving a Trigonometric Equation in Real Life AUTOMOTIVE ENGINEERING The heights h (in inches) of pistons 1 and 2 in an automobile engine can be modeled by 4π 3 1 2 3 4 5 6 h 1 = 3.75 sin 733t + 7.5 and h 2 = 3.75 sin 733 t + + 7.5 where t is measured in seconds. How often are these two pistons at the same height? FOCUS ON CAREERS SOLUTION Let h1 = h2 and solve for t. 4π 3 3.75 sin 733t + 7.5 = 3.75 sin 733 t + + 7.5 2932π 3 2932π 3 sin 733t = sin 733t cos + cos 733t sin 12 32 sin 733t = (sin 733t) º + (cos 733t) º 3 3 sin 733t = º cos 733t 2 2 3 3 tan 733t = º RE FE L AL I AUTO MECHANIC INT Auto mechanics inspect and repair mechanical and electrical systems of motor vehicles. They may specialize in areas such as transmission systems or diagnostic services. NE ER T CAREER LINK www.mcdougallittell.com 3 733t = tanº1 º + nπ 3 π 6 π nπ t = º + 4398 733 733t = º + nπ π 733 The heights are equal once every second. So in one second, π 733 733 π the heights are equal the following number of times: 1 = ≈ 233.3. 14.6 Using Sum and Difference Formulas 871 Page 4 of 6 GUIDED PRACTICE ✓ Concept Check ✓ Vocabulary Check 1. Give the sum and difference formulas for sine, cosine, and tangent. 2. Fill in the blanks for each of the following equations. ? cos 30° º cos 45° sin ? a. sin (45° º 30°) = sin ? + tan 60° b. tan (90° + 60°) = ? 1 º tan 90° 3. Explain how you can evaluate tan 105° using either the sum or difference formula for tangent. Skill Check ✓ Find the exact value of the expression. 4. cos 105° 5. sin 15° 6. tan 75° 11π 7. cos 12 23π 8. sin 12 7π 9. tan 12 Solve the equation for 0 ≤ x < 2π. 4π π 13. sin x º = 2 sin x º 3 3 π π 10. 2 sin x + = tan 3 3 π π 11. tan x + = tan x + 6 4 π π 12. cos x º = 1 + cos x + 6 6 π 14. 4 sin (x + π) = 2 cos x + + 2 2 16. 15. ºcos x = 1 + 2 cos (x º π) AUTOMOTIVE ENGINEERING Look back at Example 6 on page 871. The height h3 of piston 3 in the same engine can be modeled by 2π h3 = 3.75 sin 733 t + + 7.5 3 where t is measured in seconds. How often is piston 3 the same height as piston 2? PRACTICE AND APPLICATIONS STUDENT HELP FINDING VALUES Find the exact value of the expression. Extra Practice to help you master skills is on p. 960. 17. cos 210° 18. tan 195° 19. tan 225° 20. sin (º15°) 21. cos (º225°) 22. sin 165° 11π 23. tan 12 17π 24. cos 12 11π 25. sin º 12 π 26. cos 12 5π 27. tan º 12 5π 28. sin 12 STUDENT HELP HOMEWORK HELP Example 1: Exs. 17–28 Example 2: Exs. 29–40 Example 3: Exs. 41–48 Example 4: Exs. 49–54 Examples 5, 6: Exs. 57–59 872 EVALUATING EXPRESSIONS Evaluate the expression given cos u = 47 with π 2 9 10 3π 2 0 < u < and sin v = º with π < v < . 29. sin (u + v) 30. cos (u + v) 31. tan (u + v) 32. sin (u º v) 33. cos (u º v) 34. tan (u º v) Chapter 14 Trigonometric Graphs, Identities, and Equations Page 5 of 6 INT STUDENT HELP NE ER T HOMEWORK HELP Visit our Web site www.mcdougallittell.com for help with problem solving in Ex. 58. EVALUATING EXPRESSIONS Evaluate the expression given that sin u = 35 with π 5 3π < u < π and cos v = º with π < v < . 2 6 2 35. sin (u + v) 36. cos (u + v) 37. tan (u + v) 38. sin (u º v) 39. cos (u º v) 40. tan (u º v) SIMPLIFYING EXPRESSIONS Simplify the expression. 41. tan (x º 2π) 42. tan (x + π) π 45. sin x º 2 3π 46. cos x + 2 43. sin (x + π) 44. cos (x + π) π 47. cos x + 2 3π 48. sin x º 2 SOLVING TRIGONOMETRIC EQUATIONS Solve the equation for 0 ≤ x < 2π. π π 51. sin x + + sin x º = 0 6 6 π π 52. cos x + + cos x º = 1 3 3 π 54. tan (x + π) + cos x + = 0 2 π π 49. cos x + º 1 = cos x º 6 6 3π 3π 50. sin x + + sin x º = 1 4 4 53. tan (x + π) + 2 sin (x + π) = 0 GEOMETRY CONNECTION In Exercises 55 and 56, use the following information. y m 2 tan †2 In the figure shown, the acute angle of intersection, †2 º †1, of two lines with slopes m1 and m2 is given by: †2 †1 m2 º m1 1 + m 1m 2 †1 tan (†2 º †1) = FOCUS ON PEOPLE m 1 tan †1 †2 x 55. Find the acute angle of intersection of the 1 lines y = x + 3 and y = 2x º 3. 2 56. Find the acute angle of intersection of the lines y = x + 2 and y = 3x + 1. 57. SOUND WAVES The pressure P of sound waves on a person’s eardrum can be modeled by a r 2πl r P = cos º 1100t where a is the maximum sound pressure (in pounds per square foot) at the source, r is the distance (in feet) from the source, l is the length (in feet) of the sound wave, and t is the time (in seconds). Simplify this formula when r = 16 feet, l = 4 feet, and a = 0.4 pound per square foot. 58. RE FE L AL I BARRIE ROKEACH has more than 20 years of experience as an aerial photographer. He has had more than one dozen individual exhibitions in museums and galleries around the country. AERIAL PHOTOGRAPHY You are at a height h taking aerial photographs. The ratio of the length WQ of the image to the length NA of the actual object is camera † WQ ƒ tan († º t) + ƒ tan t = NA h tan † where ƒ is the focal length of the camera, † is the angle with the vertical made by the line from the camera to point A and t is the tilt angle of the film. Use the difference formula for tangent to simplify the WQ NA ƒ h ratio. Then show that = when t = 0. h W œ t N A Source: Math Applied to Space Science 14.6 Using Sum and Difference Formulas 873 Page 6 of 6 59. FERRIS WHEEL The heights h (in feet) of two people in different seats on a Ferris wheel can be modeled by h1 = 28 cos 10t + 38 and π 6 h2 = 28 cos 10 t º + 38 where t is the time (in minutes). When are the two people at the same height? 60. CRITICAL THINKING You can write the sum and difference formulas for cosine as a single equation: cos (u ± v) = cos u cos v ‚ sin u sin v. Explain why the symbol ± is used on the left side, but the symbol ‚ is used on the right side. Then use the symbols ± and ‚ to write the sum and difference formulas for sine and tangent as single equations. Test Preparation MULTI-STEP PROBLEM Suppose two middle-A tuning forks are struck at 61. different times so that their vibrations are slightly out of phase. The combined pressure change P (in pascals) caused by the forks at time t (in seconds) is: P = 3 sin 880πt + 4 cos 880πt a. Graph the equation on a graphing calculator using a viewing window of 0 ≤ x ≤ 0.5 and º6 ≤ y ≤ 6. What do you observe about the graph? b. Write the given model in the form y = a cos b(x º h). c. Graph the model from part (b) to confirm that the graphs are the same. ★ Challenge VERIFYING FORMULAS Use the difference formula for cosine to verify the following formulas. 62. The sum formula for cosine, by replacing v with ºv EXTRA CHALLENGE www.mcdougallittell.com sin † 63. The difference formula for tangent, by using the identity tan † = co s † 64. The difference formula for sine, by using a cofunction identity MIXED REVIEW SIMPLIFYING EXPRESSIONS Using the given matrices, simplify the expression. (Review 4.2) A= 1 3 2 3 º1 2 0 3 1 º1 2 ,B= ,C= ,D= , E = º1 º1 º2 3 º4 1 º2 0 2 0 º2 65. AD + D 66. 3(A + C) 67. º2AB + B 68. DE + AC 69. 5CD 70. AD º CD SOLVING TRIANGLES Solve ¤ABC. (Review 13.5, 13.6) 71. A = 18°, B = 28°, b = 100 72. A = 60°, C = 95°, c = 5 73. a = 13, b = 4, c = 11 74. a = 2, b = 2.5, c = 3 C a b A c SOLVING TRIGONOMETRIC EQUATIONS Solve the equation in the interval 0 ≤ x < 2π. (Review 14.4 for 14.7) 874 =0 75. tan x + 3 76. 4 cos2 x º 3 = 0 77. 8 tan x + 8 = 0 78. º5 + 8 cos x = º1 Chapter 14 Trigonometric Graphs, Identities, and Equations B Page 1 of 8 14.7 What you should learn GOAL 1 Evaluate expressions using doubleand half-angle formulas. Use double- and half-angle formulas to solve real-life problems, such as finding the mach number for an airplane in Ex. 70. GOAL 2 Why you should learn it RE FE To model real-life situations with double- and half-angle relationships, such as kicking a football in Example 8. AL LI Using Double- and Half-Angle Formulas GOAL 1 DOUBLE- AND HALF-ANGLE FORMULAS In this lesson you will use formulas for double angles (angles of measure 2u) u 2 and half angles angles of measure . The three formulas for cos 2u below are u 2 equivalent, as are the two formulas for tan . Use whichever formula is most convenient for solving a problem. DOUBLE-ANGLE AND HALF-ANGLE FORMULAS DOUBLE-ANGLE FORMULAS cos 2u = cos2 u º sin2 u sin 2u = 2 sin u cos u cos 2u = 2 cos2 u º 1 tan 2u = 2 2 tan u 1 º tan u cos 2u = 1 º 2 sin2 u HALF-ANGLE FORMULAS sin = ± u 2 1º2cosu tan = u 2 1+c2osu tan = cos = ± u 2 u 2 1 º cos u sin u u 2 sin u 1 + cos u u 2 u 2 The signs of sin and cos depend on the quadrant in which lies. EXAMPLE 1 Evaluating Trigonometric Expressions π 8 Find the exact value of (a) tan and (b) cos 105°. SOLUTION π π a. Use the fact that is half of . 8 4 STUDENT HELP Study Tip In Example 1 note that, in u 1 general, tan ≠ tan u. 2 2 Similar statements can be made for the other trigonometric functions of double and half angles. π 8 1 π 2 4 1 º cos 1º sin 2 π 2 2 º 2 2 4 = 2 = = 2 tan = tan = º1 π 4 2 b. Use the fact that 105° is half of 210° and that cosine is negative in Quadrant II. 1 2 1+co2s210° cos 105° = cos (210°) = º 3 1 + º 2 2 º 3 2 º 3 = º = º = º 2 4 2 14.7 Using Double- and Half-Angle Formulas 875 Page 2 of 8 STUDENT HELP Study Tip 3π Because π < u < in 2 Example 2, you can multiply through the 1 inequality by to get 2 π u 3π u < < , so is in 2 2 4 2 Quadrant II. Evaluating Trigonometric Expressions EXAMPLE 2 3 5 3π 2 Given cos u = º with π < u < , find the following. u b. sin 2 a. sin 2u SOLUTION 4 a. Use a Pythagorean identity to conclude that sin u = º. 5 sin 2u = 2 sin u cos u 45 35 24 25 = 2 º º = u u b. Because is in Quadrant II, sin is positive. 2 2 u sin = 2 EXAMPLE 3 1 º co s u = 2 3 1 º º 5 4 25 = = 5 2 5 Simplifying a Trigonometric Expression cos 2† sin † + cos † Simplify . STUDENT HELP Study Tip Because there are three formulas for cos 2u, you will want to choose the one that allows you to simplify the expression in which cos 2u appears, as illustrated in Example 3. SOLUTION cos2 † º sin2 † co s 2 † = sin † + cos † sin † + cos † (cos † º sin †)(cos † + sin †) sin † + cos † EXAMPLE 4 Use a double-angle formula. = Factor difference of squares. = cos † º sin † Simplify. Verifying a Trigonometric Identity Verify the identity sin 3x = 3 sin x º 4 sin3 x. SOLUTION sin 3x = sin (2x + x) 876 Rewrite sin 3x as sin (2x + x). = sin 2x cos x + cos 2x sin x Use a sum formula. = (2 sin x cos x) cos x + (1 º 2 sin2 x) sin x Use double-angle formulas. = 2 sin x cos2 x + sin x º 2 sin3 x Multiply. = 2 sin x (1 º sin2 x) + sin x º 2 sin3 x Use a Pythagorean identity. = 2 sin x º 2 sin3 x + sin x º 2 sin3 x Distributive property = 3 sin x º 4 sin3 x Combine like terms. Chapter 14 Trigonometric Graphs, Identities, and Equations Page 3 of 8 EXAMPLE 5 INT STUDENT HELP NE ER T HOMEWORK HELP Visit our Web site www.mcdougallittell.com for extra examples. Solving a Trigonometric Equation Solve tan 2x + tan x = 0 for 0 ≤ x < 2π. SOLUTION tan 2x + tan x = 0 Write original equation. 2 tan x + tan x = 0 1 º tan2 x Use a double-angle formula. 2 tan x + tan x (1º tan2 x) = 0 Multiply each side by 1 º tan2 x. 2 tan x + tan x º tan3 x = 0 Distributive property. 3 3 tan x º tan x = 0 Combine like terms. tan x (3 º tan2 x) = 0 Factor. Set each factor equal to 0 and solve for x. tan x = 0 3 º tan2 x = 0 or 3 = tan2 x x = 0, π ±3 = tan x π 2π 4π 5π 3 3 3 3 x = , , , ✓CHECK You can use a graphing calculator to check the solutions. Graph the following function: y = tan 2x + tan x Then use the Zero feature to find the x-values for which y = 0. .......... Zero X=1.0471976 Y=0 Some equations that involve double or half angles can be solved directly—without resorting to double- or half-angle formulas. EXAMPLE 6 Solving a Trigonometric Equation x 2 Solve 2 cos + 1 = 0. SOLUTION x 2 cos + 1 = 0 2 Write original equation. x 2 2 cos = º1 x 2 Subtract 1 from each side. 1 2 cos = º Divide each side by 2. 2π x = + 2nπ 3 2 4π 3 x = + 4nπ or 4π + 2nπ 3 General solution for }} or 8π + 4nπ 3 General solution for x x 2 14.7 Using Double- and Half-Angle Formulas 877 Page 4 of 8 GOAL 2 USING TRIGONOMETRY IN REAL LIFE The path traveled by an object that is projected at an initial height of h0 feet, an initial speed of v feet per second, and an initial angle † is given by 16 v cos † x2 + (tan †)x + h0 y = º 2 2 where x and y are measured in feet. (This model neglects air resistance.) Simplifying a Trigonometric Model EXAMPLE 7 SPORTS Find the horizontal distance traveled by a football kicked from ground level (h 0 = 0) at speed v and angle †. † Not drawn to scale SOLUTION Using the model above with h 0 = 0, set y equal to 0 and solve for x. 16 v co s † 16 x º tan † = 0 (ºx) v2 c o s 2 † 16 x º tan † = 0 v2 cos2 † x2 + (tan †)x = 0 º 2 2 Let y = 0. Factor. Zero product property (Ignore ºx = 0.) 16 x = tan † v2 c o s 2 † 1 x = v2 cos2 † tan † 16 Add tan † to each side. 1 16 Multiply each side by }}v 2 cos2 †. 1 16 Use cos † tan † = sin †. 1 32 Rewrite }} as }} • 2. 1 32 Use a double-angle formula. x = v2 cos † sin † FOCUS ON 1 16 x = v2 (2 cos † sin †) APPLICATIONS x = v2 sin 2† EXAMPLE 8 1 32 Using a Trigonometric Model SPORTS You are kicking a football from ground level with an initial speed of 80 feet per second. Can you make the ball travel 200 feet? RE FE L AL I FIELD GOALS The SOLUTION 1 32 200 = (80)2 sin 2† INT longest professional field goal was 63 yards, made by Tom Dempsy in 1970. This record was tied by Jason Elam during the 1998–1999 season. 1 = sin 2† NE ER T APPLICATION LINK www.mcdougallittell.com 878 Substitute for x and v in the formula from Example 7. 1 32 Divide each side by }}(80)2 = 200. 90° = 2† sinº1 1 = 90° 45° = † Solve for †. You can make the football travel 200 feet if you kick it at an angle of 45°. Chapter 14 Trigonometric Graphs, Identities, and Equations Page 5 of 8 GUIDED PRACTICE ✓ Concept Check ✓ Vocabulary Check ? formula for sine. 1. Complete this statement: sin 2u = 2 sin u cos u is called the cos 2† º cos † 2. Suppose you want to simplify . Which double-angle formula for cos † º 1 cosine would you use to rewrite cos 2†? Explain. 3. ERROR ANALYSIS Explain what is wrong in the calculations shown below. a. b. 30° tan 15° = tan 2 390° 2 1 º sin 390° 2 cos 195° = cos 1 = tan 30° 2 = 1 3 = 2 3 = 3 6 = 41 1 = 1 º 2 2 1 2 = Skill Check ✓ 2 5 3π 2 Given tan u = with π < u < , find the exact value of the expression. u 4. sin 2 u 5. cos º 2 u 6. tan 2 7. sin 2u 8. cos 2u 9. tan 2u Simplify the expression. 10. cos2 x º 2 cos 2x + 1 x 11. sin 2x tan 2 12. sin2 x + sin 2x + cos2 x tan 2x 13. s e c2 x x x 14. sin cos 2 2 15. cos 2x cot2 x º cot2 x 16. FOOTBALL Look back at Example 8 on page 878. Through what range of angles can you kick the football to make it travel at least 150 feet? PRACTICE AND APPLICATIONS STUDENT HELP Extra Practice to help you master skills is on p. 960. EVALUATING TRIGONOMETRIC EXPRESSIONS Find the exact value of the expression. 17. tan 15° 18. sin 22.5° 19. tan (º22.5°) 20. cos 67.5° 21. tan (º75°) 7π 22. cos 8 7π 23. sin º 12 5π 24. cos º 12 π 25. sin 12 u u , cos , and tan . HALF-ANGLE FORMULAS Find the exact values of sin u 2 2 2 3 π 26. cos u = , 0 < u < 5 2 2 3π 27. cos u = , < u < 2π 3 2 9 π 28. sin u = , < u < π 10 2 4 3π 29. sin u = º, < u < 2π 5 2 14.7 Using Double- and Half-Angle Formulas 879 Page 6 of 8 DOUBLE-ANGLE FORMULAS Find the exact values of sin 2x, cos 2x, and STUDENT HELP tan 2x. HOMEWORK HELP π 30. tan x = 2, 0 < x < 2 1 π 31. tan x = º, º < x < 0 2 2 1 3π 32. cos x = º, π < x < 3 2 3 3π 33. sin x = º, < x < 2π 5 2 Example 1: Exs. 17–25 Example 2: Exs. 26–33 Example 3: Exs. 34–42 Example 4: Exs. 43–50 Examples 5, 6: Exs. 51–65 Examples 7, 8: Exs. 69–73 SIMPLIFYING TRIGONOMETRIC EXPRESSIONS Rewrite the expression π without double angles or half angles, given that 0 < x < . Then simplify the 2 expression. x 34. 2 +2o csx cos 2 sin 2x 35. sin x 36. tan 2x (1 + tan x) 37. cos 2x º 3 sin2 x cos 2x 38. cos2 x sin x x 39. tan 2 1 º c o s2 x x 40. (1 + cos x) tan 2 1 + cos 2x 41. cot x sin tan 2 2 42. 1 º cos x x 2 x VERIFYING IDENTITIES Verify the identity. 43. (sin x + cos x)2 = 1 + sin 2x 44. 1 + cos 10x = 2 cos2 5x † 45. cos † + 2 sin2 = 1 2 † † 1 2† 46. sin cos = sin 3 3 2 3 47. cos 3x = cos3 x º 3 sin2 x cos x 48. sin 4† = 4 sin † cos † (1 º 2 sin2 †) 49. cos2 2x º sin2 2x = cos 4x 50. cot † + tan † = 2 csc 2† SOLVING TRIGONOMETRIC EQUATIONS Solve the equation for 0 ≤ x < 2π. 1 51. sin x = º1 2 1 52. cos x º cos x = 0 2 53. sin 2x cos x = sin x 54. cos 2x = º2 cos2 x 55. tan 2x º tan x = 0 x 56. sin + cos x = 1 2 cos 2x 57. tan 2x = 2 x 58. tan = sin x 2 cos 2x 59. =1 cos2 x FINDING GENERAL SOLUTIONS Find the general solution of the equation. 60. cos 2x = º1 61. sin 2x + sin x = 0 62. cos 2x º cos x = 0 x 63. cos º sin x = 0 2 x 64. sin + cos x = 0 2 65. cos 2x = 3 sin x + 2 66. LOGICAL REASONING Show that the three double-angle formulas for cosine are equivalent. 67. LOGICAL REASONING Show that the two half-angle formulas for tangent are equivalent. 68. Writing Use the formula at the top of page 878 to explain why the projection angle that maximizes the distance a projectile travels is † = 45° when h0 = 0. 69. PROJECTILE HEIGHT Find a formula for the maximum height of an object projected from ground level at speed v and angle †. To do this, find half of the 1 32 horizontal distance v2 sin 2† and then substitute it for x in the general model for the path of a projectile (where h0 = 0) at the top of page 878. 880 Chapter 14 Trigonometric Graphs, Identities, and Equations Page 7 of 8 FOCUS ON APPLICATIONS 70. AERONAUTICS An airplane’s mach number M is the ratio of its speed to the speed of sound. When an airplane travels faster than the speed of sound, the sound waves form a cone behind the airplane (see Lesson 10.5, page 620). † 1 The mach number is related to the apex angle † of the cone by sin = . 2 M Find the angle † that corresponds to a mach number of 4.5. INCA DWELLING In Exercises 71 and 72, use the following information. Shown below is a drawing of an Inca dwelling found in Machu Picchu, about 50 miles northwest of Cuzco, Peru. All that remains of the ancient city today are stone ruins. RE FE L AL I 71. Express the area of the triangular portion of the side of the dwelling as a function of MACHU PICCHU is an ancient Incan city discovered by Hiram Bingham in 1911 and located in the Andes Mountains near Cuzco, Peru. The site consists of 5 square miles of terraced gardens linked by 3000 steps. 18 ft 18 ft † † † sin and cos . 2 2 72. Express the area found in Exercise 71 as a function of sin †. Then solve for † assuming that the area is 132 square feet. 73. OPTICS The index of refraction n of a transparent material is the ratio of the speed of light in a vacuum to the speed of light in the material. Some common materials and their indices are air (1.00), water (1.33), and glass (1.5). Triangular prisms are often used to measure the index of refraction based on this formula: † å sin + 2 2 n = † sin å air † 2 t ligh For the prism shown, å = 60°. Write the index † 2 of refraction as a function of cot . Then find † prism if the prism is made of glass. Test Preparation QUANTITATIVE COMPARISON In Exercises 74–76, choose the statement that is true about the given quantities. A ¡ B ¡ C ¡ D ¡ The quantity in column A is greater. The quantity in column B is greater. The two quantities are equal. The relationship cannot be determined from the given information. Column A Column B 74. sinsin x, x, with 45°45° ≤< x≤ 90°90° sin 2x with x< sin 2x 75. coscos x, x, with 90°90° ≤< x≤ 135° with x< 135° cos 2x cos 2x 76. tantan x, x, with 45°45° ≤< x≤ 90°90° tan 2x with x< tan 2x ★ Challenge 1 77. DERIVING FORMULAS Use the diagram shown at the right to derive the formulas for EXTRA CHALLENGE † 2 † 2 † 2 † 2 sin , cos , and tan when † is an acute angle. † cos † 1 sin † www.mcdougallittell.com 14.7 Using Double- and Half-Angle Formulas 881 Page 8 of 8 MIXED REVIEW FUNCTION OPERATIONS Let ƒ(x) = 4x + 1 and g(x) = 6x. Perform the indicated operation and state the domain. (Review 7.3) 78. ƒ(x) + g(x) 79. ƒ(x) º g(x) 80. ƒ(x) • g(x) 81. ƒ(x) ÷ g(x) 82. ƒ(g(x)) 83. g(ƒ(x)) CALCULATING PROBABILITIES Calculate the probability of rolling a die 30 times and getting the given number of 4’s. (Review 12.6) 84. 1 85. 3 86. 5 87. 6 88. 8 89. 10 SIMPLIFYING EXPRESSIONS Simplify the expression. (Review 14.6) 3π 90. cos x º 2 91. sin (x º π) 96. π 94. sin x + 2 93. cos (x º π) π 95. tan x + 4 π 92. tan x º 3 MOVING The truck you have rented for moving your furniture has a ramp. If the ramp is 20 feet long and the back of the truck is 3 feet above the ground, at what angle does the ramp meet the ground? (Review 13.4) QUIZ 3 Self-Test for Lessons 14.6 and 14.7 Find the exact value of the expression. (Lesson 14.6) 1. sin 105° 2. cos 285° 3. tan 165° 17π 4. sin 12 13π 5. cos 12 π 6. tan º 12 1 π Given sin u = with < u < π, find the exact value of the expression. 3 2 (Lesson 14.7) u 7. sin 2 u 8. cos 2 10. sin 2u 11. cos 2u u 3+ 2 2 9. tan 2 3 º2 2 12. tan 2u Simplify the expression. (Lessons 14.6, 14.7) 13. sin (x + 3π) 14. cos (π º x) sin 2x 16. 2 co s x x 17. 2cos2 º cos x 2 1 º tan x 18. tan 2x 2 π 15. tan x + 4 Solve the equation. (Lessons 14.6, 14.7) 19. sin 3x = 0.5 20. cos 2x º cos2 x = 0 21. sin (2x º π) = sin x 22. tan (º2x) = 1 23. 1 GOLF Use the formula x = v2 sin 2† to find the horizontal distance x 32 (in feet) that a golf ball will travel when it is hit at an initial speed of 50 feet per second and at an angle of 40°. (Lesson 14.7) 882 Chapter 14 Trigonometric Graphs, Identities, and Equations Page 1 of 5 CHAPTER 14 Chapter Summary WHAT did you learn? WHY did you learn it? Graph sine, cosine, and tangent functions. (14.1) Graph the height of a boat moving over waves. (p. 836) Graph translations and reflections of sine, cosine, and tangent graphs. (14.2) Graph the height of a person rappelling down a cliff. Use trigonometric identities to simplify expressions. (14.3) Simplify the parametric equations that describe a carousel’s motion. (p. 854) Verify identities that involve trigonometric expressions. (14.3) Show that two equations modeling the shadow of a sundial are equivalent. (p. 853) Solve trigonometric equations. (14.4, 14.6, 14.7) Solve an equation that models the position of the sun at sunrise. (p. 860) Write sine and cosine models for graphs and data. Write models for temperatures inside and outside an igloo. (p. 866) (14.5) (p. 843) Use sum and difference formulas. (14.6) Relate the length of an image to the length of an actual object when taking aerial photographs. (p. 873) Use double- and half-angle formulas. (14.7) Find the angle at which you should kick a football to make it travel a certain distance. (p. 878) Use trigonometric functions to solve real-life problems. (14.1–14.7) Model real-life patterns, such as the vibrations of a tuning fork. (p. 833) How does Chapter 14 fit into the BIGGER PICTURE of algebra? In Chapter 14 you continued your study of trigonometry, focusing more on algebra connections than geometry connections. You graphed trigonometric functions and studied characteristics of the graphs, just as you have done with other types of functions during this course. In this chapter you saw how some algebraic skills are used in trigonometry, such as in solving a trigonometric equation in quadratic form. If you go on to study higher-level algebra, you will see how trigonometry is used in algebra, such as in finding the complex nth roots of a real number. STUDY STRATEGY How did you use multiple methods? Here is an example of two methods used for Example 2 on page 849 following the Study Strategy on page 830. Multiple Methods (1) sec † tan2 † + sec † = sec † (sec2 † º 1) + sec † = sec3 † º sec † + sec = sec3 † (2) sec † tan2 † + sec † = sec † † (tan2 † + 1) = sec † (sec2 †) = sec3 † 883 Page 2 of 5 CHAPTER 14 Chapter Review VOCABULARY • periodic function, p. 831 • cycle, p. 831 14.1 • period, p. 831 • amplitude, p. 831 • frequency, p. 833 • trigonometric identities, p. 848 Examples on pp. 831–834 GRAPHING SINE, COSINE, AND TANGENT FUNCTIONS EXAMPLES You can graph a trigonometric function by identifying the characteristics and key points of the graph. 1 2 y = tan 3x y = 2 sin 4x π 2 2π |4| Period = = Amplitude = |2| = 2 π |3| π4 π2 π8, 2 38π, º2 Intercepts: (0, 0); , 0 ; , 0 Maximum: π 3 Period = = Intercept: (0, 0) π 6 π 6 Asymptotes: x = , x = º 1π2 12 Minimum: 1 2 π 12 Halfway points: , ; º, º 1 1 x π 8 π π 6 6 x Draw one cycle of the function’s graph. 1 1. y = sin x 4 14.2 1 2. y = cos πx 2 3. y = tan 2πx 2 4. y = 3 tan x 3 Examples on pp. 840–843 TRANSLATIONS AND REFLECTIONS OF TRIGONOMETRIC GRAPHS EXAMPLE π 4 To graph y = 2 º 4 cos 2 x + , start with the graph of y = 4 cos 2x. π Translate the graph left units and up 2 units, and reflect it in the line y = 2. 4 2π Amplitude = |º4| = 4 Period = = π 6 |2| π On y = 2: (0, 2); , 2 2 π4 Maximum: , 6 884 π 4 34π Minimums: º, º2 ; , º2 Chapter 14 Trigonometric Graphs, Identities, and Equations π 2 x Page 3 of 5 Graph the function. 5. y = 5 sin (2x + π) 14.3 1 7. y = 2 + tan x + π 2 6. y = º4 cos (x º π) Examples on pp. 848–851 VERIFYING TRIGONOMETRIC IDENTITIES EXAMPLE You can verify identities such as sin x + cot x cos x = csc x. csoins xx sin x + cot x cos x = sin x + cos x Reciprocal identity sin2 x + cos2 x sin x Write as one fraction. = 1 sin x Pythagorean identity = csc x Reciprocal identity = Simplify the expression. π 9. csc2 (ºx) cos2 º x 2 8. tan (ºx) cos (ºx) π 10. sin2 º x º 2 sin2 x + 1 2 Verify the identity. tan2 x 11. sin2 (ºx) = tan2 x + 1 14.4 12. 1 º cos2 x = tan2 (ºx) cos2 x Examples on pp. 855–858 SOLVING TRIGONOMETRIC EQUATIONS You can find the general solution of a trigonometric equation or just the solution(s) in an interval. EXAMPLE 3 tan2 x º 1 = 0 Write original equation. 3 tan2 x = 1 Add 1 to each side. 1 3 tan2 x = Divide each side by 3. 3 3 tan x = ± Take square roots of each side. π 6 5π 6 There are two solutions in the interval 0 ≤ x < π: x = and x = . The general π 6 5π 6 solution of the equation is: x = + nπ or x = + nπ where n is any integer. Find the general solution of the equation. 13. 2 sin2 x tan x = tan x 14. sec2 x º 2 = 0 15. cos 2x + 2 sin2 x º sin x = 0 16. tan2 3x = 3 17. 2 sin x º 1 = 0 18. sin x (sin x + 1) = 0 Chapter Review 885 Page 4 of 5 14.5 Examples on pp. 862–864 MODELING WITH TRIGONOMETRIC FUNCTIONS 34π , 4 EXAMPLE You can write a model for the sinusoid at the right. Since the maximum and minimum values of the function do not occur at points equidistant from the x-axis, the curve has a vertical shift. To find the value of k, add the maximum and minimum values and divide by 2. 4 + (º2) 2 M+m 2 1 π 4 2 2 k = = = = 1 2π b x π4 , 2 π 4 The period is = π, so b = 2. Because the minimum occurs at , the graph is a sine curve that involves a reflection but no horizontal shift. The amplitude is Mºm 2 4 º (º2) 2 |a| = = = 3. Since a < 0, a = º3. The model is y = 1 º 3 sin 2x. Write a trigonometric function for the sinusoid with maximum at A and minimum at B. π 3π π 20. A(0, 6), B(2π, 0) 21. A(0, 1), B , º1 19. A , 2 , B , º2 2 2 2 14.6 Examples on pp. 869–871 USING SUM AND DIFFERENCE FORMULAS EXAMPLE You can use formulas to evaluate trigonometric functions of the sum or difference of two angles. sin 105° = sin (45° + 60°) = sin 45° cos 60° + cos 45° sin 60° 2 1 2 3 2 + 6 = 2 •2+ 2 • 2 = 4 Find the exact value of the expression. 22. sin 150° 14.7 23. cos 195° 7π 25. tan 12 24. tan 15° 13π 26. cos 12 Examples on pp. 875–878 USING DOUBLE- AND HALF-ANGLE FORMULAS EXAMPLE You can use formulas to evaluate some trigonometric functions. 3 π 1 º cos 1 º 2 6 1 π π = = 2 º 3 tan = tan • = π 2 6 12 1 sin 6 2 Find the exact value of the expression. 27. tan 165° 886 28. sin 67.5° 5π 29. cos 8 π 30. cos 12 Chapter 14 Trigonometric Graphs, Identities, and Equations 31. sin 6π Page 5 of 5 Chapter Test CHAPTER 14 Draw one cycle of the function’s graph. 1 1. y = 3 cos x 4 1 2. y = 4 sin πx 2 5 3. y = tan x 2 4. y = º2 tan 2x 5. y = º3 + 2 cos (x º π) 6. y = 1 º cos x 1 7. y = 5 + sin x 2 8. y = 5 + 2 tan (x + π) Simplify the expression. cos 2x + sin2 x 10. cos2 x π 9. cos x º 2 tan 2x sec2 x 11. º 2 tan x 1 º tan2 x 4 sin x cos x º 2 sin x sec x 12. 2 tan x Verify the identity. x 14. tan = csc x º cot x 2 13. º2 cos2 x tan (ºx) = sin 2x 15. cos 3x = cos3 x º 3 sin2 x cos x Solve the equation in the interval 0 ≤ x < 2π. Check your solutions. 17. tan2 x º 2 tan x + 1 = 0 16. º6 + 10 cos x = º1 18. tan (x + π) + 2 sin (x + π) = 0 Find the general solution of the equation. 19. 4 º 3 sec2 x = 0 21. cos x csc2 x + 3 cos x = 7 cos x 20. cos x º sin x sin 2x = 0 Find the exact value of the expression. 22. sin 345° 23. tan 112.5° 13π 25. tan 12 24. cos 375° π 26. sin 8 41π 27. cos 12 Find the amplitude and period of the graph. Then write a trigonometric function for the graph. 28. 29. 2 30. 2 3 x π 6 x 1 x π 2 31. TIDES The depth of the ocean at a swim buoy reaches a maximum of 6 feet at 3 A.M. and a minimum of 2 feet at 9 A.M. Write a trigonometric function that models the water depth y (in feet) as a function of time t (in hours). Assume that t = 0 represents 12:00 A.M. 32. TEMPERATURES The average daily temperature T (in degrees Fahrenheit) in Baltimore, Maryland, is given in the table. The variable t is measured in months, with t = 0 representing January 1. Use a graphing calculator to write a trigonometric model for T as a function of t. Source: U.S. National Oceanic and Atmospheric Administration t 0.5 1.5 2.5 3.5 4.5 5.5 6.5 7.5 8.5 9.5 10.5 11.5 T 75 79 87 94 98 101 104 105 100 92 87 77 Chapter Test 887 Page 1 of 2 Chapter Standardized Test CHAPTER 14 TEST-TAKING STRATEGY Long-term preparation for the SAT can be done throughout your high school career and can improve your overall abilities. If you keep up with your homework, both your problem-solving abilities and your vocabulary will improve. This type of long-term preparation will definitely affect not only your SAT scores, but your overall future academic performance as well. 1. MULTIPLE CHOICE Which function is graphed? 1 π A ¡ C ¡ E ¡ x π B ¡ D ¡ y = tan 2x 1 y = tan x 2 5. MULTIPLE CHOICE What is the exact value of 5π tan ? 12 A ¡ D ¡ 3 º 23 B ¡ E ¡ A ¡ tan † = 3 4 B ¡ csc † = C ¡ sec † = 5 3 D ¡ tan † = º E ¡ cot † = 2 + 3 2 º 3 C ¡ 3 º 1 23 + 3 3 6. MULTIPLE CHOICE Given that cos † = º and 5 3π π < † < , which of the following is true? 2 y = ºtan 2x 1 y = ºtan x 2 y = º2 tan x 2. MULTIPLE CHOICE Which function is graphed? 5 4 4 3 3 4 7. MULTIPLE CHOICE What trigonometric function has a graph with maximum (π, 2) and minimum (3π, º2)? 2 x π 4 A ¡ C ¡ E ¡ B ¡ D ¡ 1 4 y = 4 cos x y = 3 + 4 cos 4x 1 4 y = 3 + 4 cos x y = 4 cos 4x y = 4 + 3 cos 4x 3. MULTIPLE CHOICE What is the simplified form of π sin2 º x tan2 x + cos2 (ºx) + tan2 x? 2 A ¡ D ¡ sec2 x 2 1 + sec x B ¡ E ¡ 2 tan2 x 2 C ¡ csc2 x 2 sin x + tan x 4. MULTIPLE CHOICE Which of the following is a solution of the equation tan2 x cos x = cos x? ¡ x=0 D ¡ x = A 5π 6 ¡ x = E ¡ x = B π 6 5π 4 ¡ C π 2 x = A ¡ C ¡ E ¡ 1 2 y = 2 sin x y = 2 cos 2x 1 2 y = 2 cos x π 2 y = 2 sin x 8. MULTIPLE CHOICE Which of the following is a tan x sin 2x + 2 sin2 x solution of the equation = 1? º1 + 4 sin x A ¡ D ¡ x=0 x=π B ¡ E ¡ 5π 6 x = C ¡ 11π 6 x = There is no solution. 9. MULTIPLE CHOICE What is the simplified form of x 2 sin x tan 2 the expression ? π cos º x sin (ºx) + 1 2 A ¡ B ¡ 2 sec2 x ¡ D ¡ 2 cos2 x 1 º co s x cos2 x 2 º 2 cos x C cos2 x 2 sin x E cos2 x ¡ 888 B ¡ D ¡ y = 2 sin 2x Chapter 14 Trigonometric Graphs, Identities, and Equations Page 2 of 2 QUANTITATIVE COMPARISON In Exercises 10 and 11, choose the statement that is true about the given quantities. A ¡ B ¡ C ¡ D ¡ The quantity in column A is greater. The quantity in column B is greater. The two quantities are equal. The relationship cannot be determined from the given information. Column A Column B 10. Amplitude of the graph of y = º5 + 4 sin 3πx Amplitude of the graph of y = 3 º 5 sin 2πx 11. Period of the graph of y = tan 4πx Period of the graph of y = 3 tan 4x 12. MULTI-STEP PROBLEM The average daily time R of the sunrise and the average daily time S of the sunset for each month in Dallas, Texas, is given in the table. The variable t is measured in months, with t = 0 representing January 1. t 0.5 1.5 2.5 3.5 4.5 5.5 6.5 7.5 8.5 9.5 10.5 11.5 R 7:29 7:10 6:37 5:58 5:29 5:20 5:31 5:51 6:11 6:32 6:58 7:21 S 17:45 18:12 18:36 18:58 19:20 19:36 19:35 19:11 18:33 17:54 17:27 17:24 a. Use a graphing calculator to find trigonometric models for R and S as functions of t. When entering the data into the calculator, you must convert the number of minutes into a fraction of an hour. For example, enter 7:27 as 7 + (27 / 60). b. Graph the functions you found in part (a). Use a viewing window of 0 ≤ x ≤ 48 and 0 ≤ y ≤ 24. Describe the periods, amplitudes, and locations of local maximums and minimums. How are the functions alike? How are they different? c. Let D = S º R. What does D represent? d. Graph D in the same viewing window as R and S. How are the maximums and minimums of the three functions related? Explain the real-life significance of the relationships. 13. MULTI-STEP PROBLEM The average number of daylight hours DM in Great Falls, Michigan, is given in the table. The variable t is measured in months, with t = 0 representing January 1. t 0.5 1.5 2.5 3.5 4.5 5.5 6.5 7.5 8.5 9.5 10.5 11.5 DM 8:57 10:18 11:55 13:38 15:07 15:54 15:31 14:14 12:34 10:51 9:21 8:32 a. Use a calculator to find a trigonometric model for DM as a function of t. b. Use the table given in Exercise 12. Subtract each R-value from its corresponding S-value to find the average number of hours of sunlight a day for each month in Dallas. Use a graphing calculator to find a trigonometric model for the data as a function of t. c. Graph the functions you found in parts (a) and (b). Use a viewing window of 0 ≤ x ≤ 48 and 0 ≤ y ≤ 24. Describe the periods, amplitudes, and locations of local maximums and minimums of the functions. How are the functions alike? How are they different? Do the graphs intersect? If so, where? Chapter Standardized Test 889 Page 1 of 2 CHAPTER 14 Cumulative Practice for Chapters 1–14 Write an equation of the line with the given characteristics. (2.4) 1. slope: º2, y-intercept: 7 2. points: (5, 0), (º3, 2) 3. vertical line through (4, 2) Solve the system. (3.1, 3.2, 3.6, 4.3, 4.5, 10.7) 4. x º 2y = 6 3x + y = 4 x2 + y2 º 6x º 8y + 16 = 0 ºx + 2y º z = 2 3x º y + 4z = 10 Solve the matrix equation. (4.4) 7. 6. x2 + y2 = 16 5. x + y + z = 10 4 3 2 º5 X= º1 º1 3 º1 8. 5 3 º1 6 X= 7 4 2 0 9. 8 º1 6 0 X= º2 0 3 º2 Perform the indicated operations. (6.3, 6.5, 9.4, 9.5) 10. (º2x2 º x + 4) º (3x + 10) 11. (x º 4)(2x2 + 3x º 1) 12. (x3 º 5x + 6) ÷ (x º 2) x2 º 36 x+6 13. ÷ 2x 8x + 10 6x xº4 + 14. xº2 x2 + 3x º 10 4x 1 15. º xº7 x+7 Evaluate the expression without using a calculator. (7.1, 8.4) 5 4 16. 82/3 17. 125º1/3 18. º93/2 19. º 1 20. 1 0,0 00 1 21. log2 16 22. log3 81 23. ln e7 24. log 0.01 25. log5 1 Find the distance between the two points. Then find the midpoint of the line segment connecting the two points. (10.1) 26. (0, 0), (3, º8) 27. (º5, 0), (0, 2) 28. (º1, º4), (2, 3) 29. (7, 4), (0, º3) Write the next term of the sequence. Then write a rule for the nth term. (11.1º11.3) 30. 1, 4, 9, 16, . . . 31. 8, 4, 2, 1, . . . 32. 2, 6, 18, 54, . . . 33. º6, º1, 4, 9, . . . Find the sum of the series. (11.1º11.4) 10 34. 5 ∑ 16 35. i=1 ∑ 100012 4 ∑ (3i º 1) 36. i=1 ∑ 2º13 ‡ i 37. i=0 nº1 n=1 Find the number of permutations or combinations. (12.1, 12.2) 38. 6P5 39. 10P2 40. 3P3 41. 8C1 42. 4C2 43. 7C4 Find the arc length and area of a sector with the given radius r and central angle †. (13.2) 44. r = 11 cm, † = 80° 45. r = 6 in., † = 270° 46. r = 3 ft, † = 120° Evaluate the function without using a calculator. (13.3) 47. tan 390° 890 48. sin (º45°) Chapters 1–14 49. csc 90° 3π 50. cot º 4 5π 51. cos 3 Page 2 of 2 Evaluate the expression without using a calculator. Give your answer in both radians and degrees. (13.4) 52. cosº1 0 1 53. sinº1 2 54. tanº1 1 2 55. cosº1 º 2 56. tanº1 (º3 ) Solve ¤ABC. (13.5, 13.6) 57. A = 65°, a = 7, b = 4 58. B = 110°, a = 3, c = 8 59. a = 10, b = 9, c = 4 61. C = 98°, a = 34, b = 20 62. a = 7, b = 4, c = 6 Find the area of ¤ABC. (13.5, 13.6) 60. A = 63°, c = 13, b = 20 Graph the parametric equations. Then write an xy-equation and state the domain. (13.7) 1 63. x = t + 1, y = t º 3 for 0 ≤ t ≤ 4 4 64. x = º2t, y = t + 3 for 1 ≤ t ≤ 5 Graph the function. (14.1, 14.2) 1 66. y = 4 sin πx 3 65. y = 5 cos 2x Simplify the expression. (14.3) 2 69. tan (ºx) + tan x sec x 67. y = 5 + sin 4x π sin º x 2 70. sin x 1 68. y = º3 + tan x 2 71. tan x sec x º csc x sec2 x Find the general solution of the equation. (14.4) + 5 sin x 72. 3 sin x = 3 x 73. 2 cos2 º 1 = 0 2 74. cos x sin2 x º cos x = 0 Find the exact value of the expression. (14.6, 14.7) 75. sin 255° 76. sin 157.5° 77. tan 105° π 78. tan 12 13π 79. cos 12 80. FRACTAL GEOMETRY Tell whether c = 1 + i is in the Mandelbrot set. Use absolute value to justify your answer. (5.4) 81. GIRLS BASKETBALL The heights (in inches) of the girls chosen for the first team on PARADE’s 23rd annual All-America High School Girls Basketball Team are listed below. Find the mean, median, mode(s), range, and standard deviation of the heights. Draw a box-and-whisker plot for the heights. Source: Parade Magazine (7.7) 76, 74, 76, 71, 72, 78, 66, 68, 74, 69 82. EQUAL GENDERS What is the probability that a family with four children has exactly two girls and two boys in any order? Assume that having a girl and having a boy are equally likely events. (12.6) 83. RIALTO TOWER Suppose you are looking at the Rialto Tower in Melbourne, Australia, which reaches a height of 794 feet. Your angle of elevation to the top of the building is 39.8°. How far are you from the base of the building? Source: Council on Tall Buildings and Urban Habitat (13.1) 84. BICYCLING As you pedal up a hill, the pedals on your mountain bike make one revolution every two seconds. The maximum height of the pedal is 19 inches above the ground and the minimum height is 5 inches above the ground. Write a trigonometric model for the height H of the pedal as a function of time t. (14.1, 14.2) Cumulative Practice 891 Page 1 of 2 PROJECT Applying Chapters 13 and 14 The Mathematics of Music OBJECTIVE Explore the relationship between music and trigonometric functions. Materials: plastic or glass bottle, container of water, CBL, CBL microphone, TI-82 or TI-83 graphing calculator with cable to link to the CBL INT STUDENT HELP NE ER T TECHNOLOGY HELP For suggestions about doing this project see www.mcdougallittell.com Sound is a variation in pressure transmitted through air, water, or other matter. Sound travels as a wave. The sound of a pure note can be represented using a sine wave (or a cosine wave; recall that a cosine wave is just a sine wave shifted horizontally). More complicated sounds can be modeled by the sum of several sine waves. The pitch of a sound wave is determined by the wave’s frequency. The greater the frequency, the higher the pitch. Note Frequency (cycles/second) middle C D E F G A B C 262 294 330 349 392 440 494 523 INVESTIGATION 1. Fill a 12–20 ounce plastic or glass bottle almost to the top with water. Blow across the top and listen to the note produced. Pour a small amount of water out and repeat. Continue to remove water and blow notes until the bottle is empty. What happens to the frequency of the notes as the water level decreases? How can you tell? 2. Fill the bottle partway with water and blow across the top to create a note with constant pitch and volume. If you have trouble producing a steady stream of air, use a straw to blow across the bottle top. Use the CBL and the CBL microphone to collect the sound data and store it in the graphing calculator. Use the graphing calculator to graph the pressure of the sound as a function of time. The graph should resemble a sine wave. 3. Use the graph of the sound data to calculate the frequency of the note—the number of complete cycles in one second. 4. Write a sine function to describe the note. 5. Choose a note from the table. Try producing the note as follows: Adjust the water level in the bottle, blow across the top, and use the CBL and graphing calculator to find the frequency of the resulting note. Repeat this process until the frequency of the note you produce is approximately equal to the frequency of the note you chose from the table. 6. Write a sine function to describe the note you chose from the table. 892 Chapters 13 and 14 Page 2 of 2 PRESENT YOUR RESULTS Write a report to present your results. • • Explain how you used the CBL. Explain how you found the frequency of a note from the note’s sine wave. • Explain how you wrote the sine functions in Exercises 4 and 6. • Include a sketch of the water level in your bottle for both notes you produced in Exercises 2 and 5. • Include a graph of the sine wave for each of the two notes. • Consider including a recording of the notes you produced. You might repeat the experiment to produce several different notes. • Describe how you used your knowledge of trigonometric functions in this project. EXTENSION Choose a note to play and have a classmate also choose a note. Find sine functions y = ƒ(x) and y = g(x) that model the two notes (as you did in Exercises 4 and 6 of the investigation). Then play the notes simultaneously and use the CBL and graphing calculator to graph the resulting sound wave. Compare this graph with the graph of y = ƒ(x) + g(x). What do you notice? MUSIC CONNECTION Bring in musical instruments and play a pure note on each. Use the CBL and graphing calculator to find the sine function that corresponds to each note. Observe what happens when you change the volume of the notes. Also, compare the sine waves for different instruments playing the same note. Project 893