2.R: Periodic Functions (Review)
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- May 5, 2023
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6.1: Graphs of the Sine and Cosine Functions
For the exercises 1-8, graph the functions for two periods and determine the amplitude or stretching factor, period, midline equation, and asymptotes.
1) f(x)=−3cosx+3
- Answer
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amplitude: 3; period: 2π; midline: y=3
no asymptotes;
2) f(x)=14sinx
3) f(x)=3cos(x+π6)
- Answer
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amplitude: 3; period: 2π; midline: y=0; no asymptotes
4) f(x)=−2sin(x−2π3)
5) f(x)=3sin(x−π4)−4
- Answer
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amplitude: 3; period: 2π; midline: y=−4; no asymptotes
6) f(x)=2(cos(x−4π3)+1)
7) f(x)=6sin(3x−π6)−1
- Answer
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amplitude: 6; period: dfrac2π3; midline: y=−1; no asymptotes
8) f(x)=−100sin(50x−20)
6.2: Graphs of the Other Trigonometric Functions
For the exercises 1-4, graph the functions for two periods and determine the amplitude or stretching factor, period, midline equation, and asymptotes.
1) f(x)=tanx−4
- Answer
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stretching factor: none; period: π
midline: y=−4; asymptotes: x=π2+πk, where k is an integer;
2) f(x)=2tan(x−π6)
3) f(x)=−3tan(4x)−2
- Answer
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stretching factor: 3; period: π4; midline: y=−2; asymptotes: x=π8+π4k, where k is an integer
4) f(x)=0.2cos(0.1x)+0.3
For the exercises 5-10, graph two full periods. Identify the period, the phase shift, the amplitude, and asymptotes.
5) f(x)=13secx
- Answer
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amplitude: none; period: 2π; no phase shift; asymptotes: x=π2k, where k is an integer
6) f(x)=3cotx
7) f(x)=4csc(5x)
- Answer
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amplitude: none; period: 2π5; no phase shift; asymptotes: x=π5k, where k is an integer
8) f(x)=8sec(14x)
9) f(x)=23csc(12x)
- Answer
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amplitude: none; period: 4π; no phase shift; asymptotes: x=2πk, where k is an integer
10) f(x)=−csc(2x+π)
For the exercises 11-15, use this scenario: The population of a city has risen and fallen over a 20-year interval. Its population may be modeled by the following function: y=12,000+8,000sin(0.628x), where the domain is the years since 1980 and the range is the population of the city.
11) What is the largest and smallest population the city may have?
- Answer
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largest: 20,000; smallest: 4,000
12) Graph the function on the domain of [0,40].
13) What are the amplitude, period, and phase shift for the function?
- Answer
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amplitude: 8,000; period: 10; phase shift: 0
14) Over this domain, when does the population reach 18,000? 13,000?
15) What is the predicted population in 2007? 2010?
- Answer
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In 2007, the predicted population is 4,413. In 2010, the population will be 11,924.
For the exercises 16a-16d, suppose a weight is attached to a spring and bobs up and down, exhibiting symmetry.
16) Suppose the graph of the displacement function is shown in the Figure below, where the values on the x-axis represent the time in seconds and the y-axis represents the displacement in inches.
- Give the equation that models the vertical displacement of the weight on the spring.
- At time=0, what is the displacement of the weight?
- Answer
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5 in.
- At what time does the displacement from the equilibrium point equal zero?
- What is the time required for the weight to return to its initial height of 5 inches? In other words, what is the period for the displacement function?
- Answer
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10 seconds
6.3: Inverse Trigonometric Functions
For the exercises 1-11, find the exact value without the aid of a calculator.
1) sin−1(1)
2) cos−1(√32)
- Answer
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π6
3) tan−1(−1)
4) cos−1(1√2)
- Answer
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π4
5) sin−1(−√32)
6) sin−1(cos(π6))
- Answer
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π3
7) cos−1(tan(3π4))
8) sin(sec−1(35))
- Answer
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No solution
9) cot(sin−1(35))
10) tan(cos−1(513))
- Answer
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125
11) sin(cos−1(xx+1))
12) Graph f(x)=cosx and f(x)=secx on the interval [0,2π) and explain any observations.
- Answer
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The graphs are not symmetrical with respect to the line y=x.
They are symmetrical with respect to the y-axis.
13) Graph f(x)=sinx and f(x)=cscx and explain any observations.
14) Graph the function f(x)=x1−x33!+x55!−x77! on the interval [−1,1] and compare the graph to the graph of f(x)=sinx on the same interval. Describe any observations.
- Answer
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The graphs appear to be identical.
Practice Test
For the exercises 1-13, sketch the graph of each function for two full periods. Determine the amplitude, the period, and the equation for the midline.
1) f(x)=0.5sinx
- Answer
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amplitude: 0.5; period: 2π
midline y=0; y=0
y = 0 y = 0
2) f(x)=5cosx
3) f(x)=5sinx
- Answer
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amplitude: 0.5; period: 2π; midline y=0
4) f(x)=sin(3x)
5) f(x)=−cos(x+π3)+1
- Answer
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amplitude: 1; period: 2π; midline y=1
6) f(x)=5sin(3(x−π6))+4
7) f(x)=3cos(13x−5π6)
- Answer
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amplitude: 3; period: 6π; midline y=0
8) f(x)=tan(4x)
9) f(x)=−2tan(x−7π6)+2
- Answer
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amplitude: none; period: π; midline y=0, asymptotes: x=2π3+πk
where k is an integer,
10) f(x)=πcos(3x+π)
11) f(x)=5csc(3x)
- Answer
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amplitude: none; period: 2π3; midline y=0, asymptotes: x=π3k
where k is an integer,
12) f(x)=πsec(π2x)
13) f(x)=2csc(x+π4)−3
- Answer
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amplitude: none; period: 2π; midline y=−3
For the exercises 14-16, determine the amplitude, period, and midline of the graph, and then find a formula for the function.
14) Give in terms of a sine function.
15) Give in terms of a sine function.
- Answer
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amplitude: 2; period: 2; midline: y=0; f(x)=2sin(π(x−1))
16) Give in terms of a tangent function.
For the exercises 17-20, find the amplitude, period, phase shift, and midline.
17) y=sin(π6x+π)−3
- Answer
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amplitude: 1; period: 12; phase shift: −6; midline: y=−3
18) y=8sin(7π6x+7π2)+6
19) The outside temperature over the course of a day can be modeled as a sinusoidal function. Suppose you know the temperature is 68∘F at midnight and the high and low temperatures during the day are 80∘F and 56∘F, respectively. Assuming t is the number of hours since midnight, find a function for the temperature, D, in terms of t.
- Answer
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D(t)=68−12sin(π12x)
20) Water is pumped into a storage bin and empties according to a periodic rate. The depth of the water is 3 feet at its lowest at 2:00 a.m. and 71 feet at its highest, which occurs every 5 hours. Write a cosine function that models the depth of the water as a function of time, and then graph the function for one period.
For the exercises 21-25, find the period and horizontal shift of each function.
21) g(x)=3tan(6x+42)
- Answer
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period: π6; horizontal shift: −7
22) n(x)=4csc(5π3x−20π3)
23) Write the equation for the graph in the Figure below in terms of the secant function and give the period and phase shift.
- Answer
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f(x)=sec(πx); period: 2; phase shift: 0
24) If tanx=3
25) If secx=4, find sec(−x).
- Answer
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4
For the exercises 26-28, graph the functions on the specified window and answer the questions.
26) Graph m(x)=sin(2x)+cos(3x) on the viewing window [−10,10] by [−3,3]
27) Graph n(x)=0.02sin(50πx) on the following domains in x:[0,1] and[0,3]
- Answer
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The views are different because the period of the wave is 125
Over a bigger domain, there will be more cycles of the graph..
28) Graph f(x)=sinxx on [−0.5,0.5] and explain any observations.
For the exercises 29-31, let f(x)=35cos(6x).
29) What is the largest possible value for f(x)?
- Answer
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35
30) What is the smallest possible value for f(x)?
31) Where is the function increasing on the interval [0,2π]?
- Answer
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On the approximate intervals (0.5,1),(1.6,2.1),(2.6,3.1),(3.7,4.2),(4.7,5.2),(5.6,6.28)
For the exercises 32-33, find and graph one period of the periodic function with the given amplitude, period, and phase shift.
32) Sine curve with amplitude 3, period π3
33) Cosine curve with amplitude 2, period \(\dfrac{\pi }{6}\)
- Answer
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f(x)=2cos(12(x+π4))+3
For the exercises 34-35, graph the function. Describe the graph and, wherever applicable, any periodic behavior, amplitude, asymptotes, or undefined points.
34) f(x)=5cos(3x)+4sin(2x)
35) f(x)=e(sint)
- Answer
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This graph is periodic with a period of 2π
For the exercises 36-43, find the exact value.
36) sin−1(√32)
37) tan−1(√3)
- Answer
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π3
38) cos−1(−√32)
39) cos−1(sin(π))
- Answer
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π2
40) cos−1(tan(7π4))
41) cos(sin−1(1−2x))
- Answer
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√1−(1−2x)2
42) cos−1(−0.4)
43) cos(tan−1(x2))
- Answer
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1√1+x4
For the exercises 44-46, suppose sint=xx+1
44) tant
45) csct
- Answer
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x+1x
46) Given Figure, find the measure of angle θ to three decimal places. Answer in radians.
For the exercises 47-49, determine whether the equation is true or false.
47) arcsin(sin(5π6))=5π6
- Answer
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False
48) arccos(cos(5π6))=5π6
49) The grade of a road is 7%. This means that for every horizontal distance of 100 feet on the road, the vertical rise is 7 feet. Find the angle the road makes with the horizontal in radians.
- Answer
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approximately 0.07 radians