5.1.2: Homework

Vocabulary Check

1. Which trigonometric functions are called the fundamental trigonometric functions?

2. The smallest distance (along the \( x \)-axis) for which a function starts a repeating pattern (that continues forever) is called the ___ of the function.

3. The ___ for a function is found by taking one-half the sum of the function's maximum and minimum values.

4. If the graph of a function is symmetric about the origin, we say that the function is ___.

Concept Check

5. List the values from \(0\) to \(2 \pi\) by multiples of \(\frac{\pi}{4}\).

6. List the values from \(0\) to \(2 \pi\) by multiples of \(\frac{\pi}{6}\).

7. List several ways in which the graph of \(y=\tan \left(\theta\right)\) is different from the graphs of \(y=\sin \left(\theta\right)\) and \(y=\cos \left(\theta\right)\).

8. Sketch two examples of a function with period 8: one that is sinusoidal and one that is not.

True or False? For the following exercises, determine if the statement is true or false. If true, cite the definition or theorem stated in the text supporting your claim. If false, explain why it is false and, if possible, correct the statement.

9. The amplitude of a function is the vertical distance between the maximum and minimum values (if these exist).

10. \(\tan \left(\theta+180^{\circ}\right)=\tan \theta\) for any value of \(\theta\).

11. If \(f(x+p)=f(x)\) for all \(x\), then the constant \(p\) is called the period of the function.

12. The tangent function has period \(\pi\). It is undefined at odd multiples of \(\frac{\pi}{2}\) and is increasing on each interval of its domain.

13. Functions that have graphs shaped like sines or cosines are called sinusoidal.

14. The cosine graph looks just like the sine graph except flipped upside down.

15. The period of a sine or cosine function is the distance between horizontal intercepts.

16. Periodic functions are used to model phenomena that exhibit cyclical behavior.

Basic Skills

17.  

    1. Use your calculator to complete the table of values. Round values to hundredths.

       \(\theta\)

       0

       \(\dfrac{\pi}{12}\)

       \(\dfrac{\pi}{6}\)

       \(\dfrac{\pi}{4}\)

       \(\dfrac{\pi}{3}\)

       \(\dfrac{5\pi}{12}\)

       \(\dfrac{\pi}{2}\)

       \(\dfrac{7\pi}{12}\)

       \(\dfrac{2\pi}{3}\)

       \(\dfrac{3\pi}{4}\)

       \(\dfrac{5\pi}{6}\)

       \(\dfrac{11\pi}{12}\)

       \(\pi\)

       \(\cos \theta\)

    2. Sketch a graph of \(y = \cos \theta\) on the grid.

       

        

18.  

    1. Use your calculator to complete the table of values. Round values to hundredths.

       \(\theta\)

       0

       \(\dfrac{\pi}{12}\)

       \(\dfrac{\pi}{6}\)

       \(\dfrac{\pi}{4}\)

       \(\dfrac{\pi}{3}\)

       \(\dfrac{\pi}{2}\)

       \(\dfrac{7\pi}{12}\)

       \(\dfrac{2\pi}{3}\)

       \(\dfrac{3\pi}{4}\)

       \(\dfrac{5\pi}{6}\)

       \(\dfrac{11\pi}{12}\)

       \(\pi\)

       \(\sin \theta\)

    2. Sketch a graph of \(y = \sin \theta\) on the grid.

       

        

19.  

    1. Sketch a graph of \(y=\cos x\) using the guide points in the table below.

       \(x\)	0	\(\dfrac{\pi}{2}\)	\(\pi\)	\(\dfrac{3\pi}{2}\)	\(2\pi\)
       \(\cos x\)

    2. State the domain and range of the function \(y=\cos x\).

20.  

    1. Sketch a graph of \(y=\tan x\) using the guide points in the table below.

       \(x\)	0	\(\dfrac{\pi}{4}\)	\(\dfrac{\pi}{2}\)	\(\dfrac{3\pi}{4}\)	\(\pi\)
       \(\cos x\)

    2. State the domain and range of the function \(y=\tan x\).

21. Sketch a graph of \(y = \sin x\) on each grid.

    

22. Sketch a graph of \(y = \cos x\) on each grid.

    

23.  

    1. Sketch a graph of \(f(x) = \sin x\) where \(x\) is a real number.

    2. State the domain and range of \(f(x) = \sin x\).

    

24.  

    1. Sketch a graph of \(g(x) = \cos x\) where \(x\) is a real number.

    2. State the domain and range of \(g(x) = \cos x\).

    

25.  

    1. Sketch a graph of \(h(x) = \tan x\) where \(x\) is a real number.

    2. State the domain and range of \(h(x) = \tan x\).

    

26. Sketch \(f(x) = \cos x\) and \(g(x) = \sin x\) on the same grid.

    

27. The graph shows your height as a function of angle as you ride the Ferris wheel. For each location A - E on the Ferris wheel, mark the corresponding point on the graph.

    

28. The graph shows your height as a function of angle as you ride the Ferris wheel. For each location F - J on the graph, mark the corresponding point on the Ferris wheel.

    

29. The graph shows the horizontal displacement of your foot from the center of the chain gear as you pedal a bicycle. For each location K - O on the chain gear, mark the corresponding point on the graph.

    

30. The graph shows the horizontal displacement of your foot from the center of the chain gear as you pedal a bicycle. For each location P - T on the graph, mark the corresponding point on the chain gear.

    

For the following exercises, use the figures below. Show your solutions on the graphs.

31.  

    1. Use the graph of \(y = \sin x\) to estimate two solutions of the equation \(\sin x = 0.65\).

    2. Use the unit circle to estimate two solutions of the equation \(\sin x = 0.35\).

32.  

    1. Use the graph of \(y = \sin x\) to estimate two solutions of the equation \(\sin x = −0.2\).

    2. Use the unit circle to estimate two solutions of the equation \(\sin x = −0.6\).

For the following exercises, use the figures below. Show your solutions on the graphs.

33.  

    1. Use the graph of \(y = \cos x\) to estimate two solutions of the equation \(\cos x = −0.4\).

    2. Use the unit circle to estimate two solutions of the equation \(\cos x = −0.8\).

34.  

    1. Use the graph of \(y = \cos x\) to estimate two solutions of the equation \(\cos x = 0.15\).

    2. Use the unit circle to estimate two solutions of the equation \(\cos x = 0.55\).

For the following exercises, use the graph of \(y = \tan x\) to estimate two solutions to the equation.

35. \(\tan x = 4\)

36. \(\tan x = 7\)

37. \(\tan x = -0.5\)

38. \(\tan x = -2.5\)

39. \(\sin x = 5.5 \cos x\)

40. \(\sin x = -4.5 \cos x\)

41. \(-2 \sin x = 3 \cos x\)

42. \(2 \sin x = \cos x\)

For the following exercises, find all solutions between \(0\) and \(2\pi\). Round to two decimal places. Sketch your solutions on a unit circle.

43. \(\cos t=0.74\)

44. \(\sin t=0.58\)

45. \(\tan t=1.6\)

46. \(\tan x=-0.6\)

47. \(\sin x=-0.72\)

48. \(\cos x=-0.48\)

Which of the graphs in the following exercises are periodic? If the graph is periodic, give its period.

49.  

    

50.  

    

51.  

    

52.  

    

53.  

    1. Is the function shown periodic? If so, what is its period? If not, explain why not.

       

        

    2. Compute the difference between the maximum and minimum function values. Sketch in the midline of the graph.

    3. Find the smallest positive value of \(k\) for which \(f(x) = f(x + k)\) for all \(x\).

    4. Find the smallest positive values of \(a\) and \(b\) for which \(f(b) − f(a)\) is a maximum.

For the following exercises, use symmetry to help in evaluating the given trigonometric function.

54. \( \cos\left( -30^{ \circ } \right) \)

55. \( \sin\left( -\frac{5\pi}{6} \right) \)

56. \( \tan\left( -135^{ \circ } \right) \)

57. \( \sec\left( -\frac{5\pi}{3} \right) \)

58. \( \csc\left( -315^{ \circ } \right) \)

59. \( \cot\left( -\frac{7\pi}{6} \right) \)

Synthesis Questions

60. An ant is walking clockwise around the face of a sundial in the garden. The sundial is a circle with a 12-inch diameter, and the ant makes one circuit of the sundial in 24 seconds.

    1. Sketch the sundial and a coordinate system with its origin at the center of the dial. Draw the \(y\)-axis to align with the gnomon (pointer) of the sundial.

    2. Suppose we start timing the ant when it is at the tip of the gnomon. Complete the table showing the ant’s location as an angle in standard position and its \(y\)-coordinate at that time.

       \(t\)	0	2	4	6	8	10	12	14	16	18	20	22	24
       \(\theta\)	\(90^{\circ}\)	\(60^{\circ}\)
       \(y=f(t)\)	\(6\)	\(3\sqrt{3}\)

    3. Sketch a graph of the ant's \(y\)-coordinate as a function of time.

       

        

    4. If the ant makes a second trip around the sundial, what will the graph look like from \(t = 24\) to \(t = 48\)? Explain the statement \(f(t + 24) = f(t)\).

61. Repeat the previous exercise, but make a table and graph of the ant’s \(x\)-coordinate as a function of time.

    \(t\)	0	2	4	6	8	10	12	14	16	18	20	22	24
    \(\theta\)	\(90^{\circ}\)	\(60^{\circ}\)
    \(x=g(t)\)	0	3

    

62. Jorge is standing at the point \((5, 0)\) on the square shown. If he walks around the square in the counter-clockwise direction, his \(y\)-coordinate is a function of the distance, \(d\), he has walked.

    

    1. What is the \(y\)-coordinate of Jorge’s position when he has walked 2 units? 5 units? 8 units? (Start filling in the table in part (c).)

    2. After Jorge reaches the upper right corner of the square, he will turn left and start walking along the top of the square. What is his \(y\)-coordinate from \(d = 5\) to \(d = 15\)?

    3. Fill in the rest of the table with the \(y\)-coordinates of Jorge’s position when he has walked a distance \(d\).

       \(d\)	0	2	5	8	10	12	15	18	20	22	25	28	30	32	35	38	40
       \(y\)

    4. Plot a graph of Jorge's coordinate as a function of the distance he has walked.

       

        

63. Repeat the previous exercise, except consider Jorge’s \(x\)-coordinate as a function of the distance he has walked.

    \(d\)	0	2	5	8	10	12	15	18	20	22	25	28	30	32	35	38	40
    \(x\)

    

    1. Refer to the previous exercise. Suppose you know where Jorge is on the square at some given time. Can you predict where he will be after he has walked another 40 units?

    2. Let \(y=f(d)\). What does your answer say about \(f(d)\) and \(f(d+40)\) for any positive value of \(d\)?

    3. How does the graph of \(f\) for \(0 \leq d \leq 40\) compare with the graph for \(40 \leq d \leq 80\)?

    4. Describe how the graph of \(f\) would continue for \(0 \leq d \leq 400\).

64. Refer to the previous exercise. Sketch a graph of Jorge's \(x\)-coordinate, \(x=g(d)\), for \(0 \leq d \leq 120\).

Proofs

For the following exercises, you may assume that sine is an odd function and cosine is an even function.

65. Prove that tangent is an odd function.

66. Prove that secant is an even function.

67. Prove that cosecant is an odd function.

68. Prove that cotangent is an odd function.

For the following exercises, prove the identity.

69. \( \cot\left( -\theta \right) \sin\left( -\theta \right) = \cos\left( \theta \right)\)

70. \( \dfrac{\cos\left( -x \right) \sec\left( x \right)}{\tan\left( -x \right)} = -\cot\left( x \right) \)

71. \( \sin\left( -y \right) \sec\left( -y \right) = \tan\left( -y \right)\)

72. \( \sin\left( -\theta \right) \sec\left( -\theta \right) \cot\left( -\theta \right) = 1 \)

73. \( 1 = 2 \cos\left( x \right) \sec\left( -x \right) + \tan\left( -y \right) \cot\left( y \right) \)

74. \( \dfrac{\tan\left( -\gamma \right)}{\sin\left( \gamma \right)} = -\sec\left( \gamma \right) \)

75. \( \tan\left( A \right) - \dfrac{1}{\tan\left( -A \right)} = \dfrac{\sec\left( -A \right)}{\sin\left( A \right)} \)

76. \( \sec\left( z \right) + \tan\left( z \right) \sin\left( -z \right) = \cos\left( -z \right) \)

77. \( \sin\left( -\phi \right) \left[ \csc\left( -\phi \right) + \sec\left( -\phi \right) \right] = 1 + \tan\left( -\phi \right)\)

Applications

For the following exercises, sketch a periodic function that models the situation.

78. At a ski slope, the lift chairs take 5 minutes to travel from the bottom, at an elevation of 3000 feet, to the top, at elevation 4000 feet. The cable supporting the ski lift chairs is a loop turning on pulleys at a constant speed. At the top and bottom, the chairs are at a constant elevation for a few seconds to allow skiers to get on and off.

    1. Sketch a graph of \(h(t)\), the height of one chair at time \(t\). Show at least two complete trips.

    2. What is the period of \(h(t)\)?

79. The heater in Wyatt’s house doesn’t have a thermostat; it runs on a timer. It uses 300 watts when it is running. Wyatt sets the heater to run from 6 am to noon, and again from 4 pm to 10 pm in the evening.

    1. Sketch a graph of \(P(t)\), the power drawn by the heater as a function of time. Show at least two days of heater use.

    2. What is the period of \(P(t)\)?

80. Piyali adds water to her fish pond once a week to keep the depth at 30 centimeters. During the week, the water evaporates at a constant rate of 0.5 centimeters per day.

    1. Sketch a graph of \(D(t)\), the depth of the water as a function of time. Show at least two weeks.

    2. What is the period of \(D(t)\)?

81. Roy’s miniature schnauzer, Sophie, is very energetic and bounces excitedly at dinner time. Sophie can jump 30 inches high, and each jump takes him 0.8 seconds.

    1. Sketch a graph of Sophie’s height, \(D(t)\), as a function of time. Show at least two jumps.

    2. What is the period of \(h(t)\)?

82. Each situation describes a periodic function. Match each situation with the appropriate graph.

    1. When the heart contracts, blood pressure in the arteries rises rapidly to a peak (systolic blood pressure) and then falls off quickly to a minimum (diastolic blood pressure). Blood pressure is a function of time.

    2. After an injection is given to a patient, the amount of the drug present in his bloodstream decreases over time. The patient receives injections at regular intervals to restore the drug level to the prescribed level. The amount of the drug present is a function of time.

    3. The monorail shuttle train between the north and south terminals at Gatwick Airport departs from the south terminal every 12 minutes. The distance from the train to the south terminal is a function of time.

    4. Ron gets a haircut every two weeks. The length of his hair is a function of time.

    

83. Match each of the following situations with an appropriate graph below.

    1. The number of hours of daylight in Salt Lake City varies from a minimum of 9.6 hours on the winter solstice to a maximum of 14.4 hours on the summer solstice.

    2. A weight is 6.5 feet above the floor, suspended from the ceiling by a spring. The weight is pulled down to 5 feet above the floor and released, rising past 6.5 feet in 0.5 seconds before attaining its maximum height of feet. Neglecting the effects of friction, the height of the weight will continue to oscillate between its minimum and maximum height.

    3. The voltage used in U.S. electrical current changes from 155V to −155V and back 60 times each second.

    4. Although the moon is spherical, what we see from earth looks like a disk, sometimes only partly visible. The percentage of the moon’s disk that is visible varies between 0 (at new moon) to 100 (at full moon).

    

84. The table shows sunrise and sunset times in Los Angeles on the fifteenth of each month.

    Month	Oct	Nov	Dec	Jan	Feb	Mar
    Sunrise	5:58	6:26	6:51	6:59	6:39	6:04
    Sunset	17:20	16:50	16:45	17:07	17:37	18:01

    Month	Apr	May	Jun	Jul	Aug	Sep
    Sunrise	5:22	4:52	4:42	4:43	5:15	5:37
    Sunset	18:25	18:48	19:07	19:05	18:40	18:00

    1. Use the left-hand grid to plot the sunrise times and sketch a sinusoidal graph through the points.

    2. Use the right-hand grid to plot the sunset times and sketch a sinusoidal graph through the points.

    

85.  

    1. Use the data from Problem 65 to complete the table with the hours of sunlight in Los Angeles on the fifteenth of each month.

       Month	Oct	Nov	Dec	Jan	Feb	Mar
       Hours of Daylight

       Month	Apr	May	Jun	Jul	Aug	Sep
       Hours of Daylight

    2. Plot the daylight hours and sketch a sinusoidal graph through the points.

    

86. Many people who believe in astrology also believe in biorhythms. The graph shows an individual’s three biorhythms, physical, emotional, and intellectual, for 36 days, from \(t = 0\) on September 30 to November 5.

    

    1. Find the dates of highest and lowest activity for each biorhythm during the month of October.

    2. Find the period of each biorhythm in days.

    3. On the day of your birth, all three biorhythms are at their maximum. How old will you be before all three are again at the maximum level?

87. The path of a satellite orbiting above the earth makes a sinusoidal graph on a map of the earth, with its midline at the equator. On the map on the next page, sketch a graph for a satellite that orbits the earth every 90 minutes, and strays no farther than 4000 km from the equator. (One degree of latitude is equal to 111 kilometers.) The satellite passes over the spot \(0^{\circ}\) latitude and \(0^{\circ}\) longitude at time \(t = 0\). Label a scale on the equator to serve as a time axis for your graph.

    

88. The graph shows arterial blood pressure, measured in millimeters of mercury (mmHg), as a function of time.

    

    1. What are the maximum (systolic) and minimum (diastolic) pressures? The pulse pressure is the difference of systolic and diastolic pressures. What is the pulse pressure?

    2. The mean arterial pressure is the diastolic pressure plus one-third of the pulse pressure. Calculate the mean arterial pressure, and draw a horizontal line on the graph at that pressure.

    3. The blood pressure graph repeats its cycle with each heartbeat. What is the heart rate, in beats per minute, of the person whose blood pressure is shown in the graph?

89. Here is a tide chart for Los Angeles for the week of December 17–23, 2000. The horizontal axis shows time in hours, with \(t = 12\) corresponding to noon on December 17. The vertical axis shows the height of the tide in feet above mean sea level.

    

    

    1. High tides occurred at 3:07 am and 2:08 pm on December 17, and low tides at 8:41 am and 9:02 pm. Estimate the heights of the high and low tides on that day.

    2. Is tide height a periodic function of time? Use the information from part (a) to justify your answer.

    3. Make a table showing approximate times and heights for the high tides throughout the week. Make a similar table for the low tides.

    4. Describe the trend in the heights of the high tides over the week. Describe the trend in the heights of the low tides.

    5. What is the largest height difference between consecutive high and low tides during the week shown? When does it occur?

90. The apparent magnitude of a star is a measure of its brightness as seen from earth. Smaller values of apparent magnitude correspond to brighter stars. The graph below, called a light curve, shows the apparent magnitude of the star Algol as a function of time. Algol is actually a system of two stars, a bright principal star and its dimmer companion, in orbit around each other. As each star passes in front of the other it eclipses some of the light that reaches earth.

    

    1. The light curve is periodic. What is its period?

    2. What is the range of apparent magnitudes of the Algol system?

    3. Explain the large and small dips in the light curve. What is happening to cause the dips?

91. Some stars, called Cepheid variable stars, appear to pulse, getting brighter and dimmer periodically. The graph shows the light curve for the star Delta Cephei.

    

    1. What is the period of the graph?

    2. What is the range of apparent magnitudes for Delta Cephei?