5.3E: Describing Relationships with The Sine and Cosine Functions (Exercises)
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Part 1: Graphs of the form \(y=A\sin(B\theta) \) or \(y=A\cos(B\theta) \)
In exercises #1 - 5, match the function with its corresponding graph given below
1. \(f\left(x\right)=3\sin \left(2x\right)\)
2. \(g\left(x\right)=\sin \left(\dfrac{1}{2}x\right)\)
3. \(h\left(x\right)=-2\cos \left(\dfrac{1}{2}x\right)\)
4. \(k\left(x\right)=-2\sin \left(\dfrac{1}{2}x\right)\)
5. \(m\left(x\right)=2\cos \left(3x\right)\)
A. 
B. 
C. 
D. 
E. 
In exercises #6 - 12, identify the amplitude and period of each function given, then graph two complete cycles with accurate scale on both axis without assistance from technology. Label at least 5 points.
6. \(f\left(x\right)=4\cos \left(3x\right)\)
7. \(g\left(x\right)=\dfrac{2}{3}\sin \left(3x\right)\)
8. \(h\left(x\right)=5\cos \left(\dfrac{\pi}{4}x\right)\)
9. \(k\left(x\right)=-8\cos \left(\dfrac{1}{3}x\right)\)
10. \(m\left(x\right)=-4\sin \left(\dfrac{\pi}{2}x\right)\)
11. \(f\left(x\right)=5\cos \left(6x\right)\)
12. \(m\left(x\right)=-2\sin \left(\dfrac{2}{5}x\right)\)
In exercises #13 - 20, identify the amplitude and period of each wave whose graph is given below, then find a possible formula for the function.
13. 
14. 
15. 
16. 
Part 2: Graphs of the form \(y=A\sin(B(\theta-h) +D\) or \(y=A\cos(B(\theta-h) +D\)
In exercises #17-24, identify the amplitude, period, phase shift, and centerline of each function given, then graph two complete cycles with accurate scale on both axis without assistance from technology. Label at least 5 points.
17. \(y=3\sin (x+\dfrac{\pi}{8}))+5\)
18. \(y=-5\cos \left(x- \dfrac{\pi}{4})\right)-7\)
19. \(y=4\sin \left(\dfrac{\pi }{2} (x-3)\right)\)
20. \(y=3\cos \left( 2 (x-\dfrac{\pi }{3})\right)+1\)
21. \(y=5\sin (5x+20)-2\)
22. \(y=\sin \left(\dfrac{\pi }{6} x+\pi \right)-3\)
23. \(y=8\cos \left(\dfrac{7\pi }{6} x+\dfrac{7\pi }{2} \right)+6\)
24. \(y=2\sin (3x-21)+4\)
In exercises #25-31, find a formula for each of the functions graphed below.
25. 
26. 
27. 
28. 
29. 
30. 
31.
Part 3: Modeling with sine and cosine.
32. The outside temperature is 50 degrees at midnight and the high and low temperature during the day are 57 and 43 degrees, respectively, over the course of two days. Assuming t is the number of hours since midnight, find a function for the temperature, D, in terms of t.
- Find a formula for a function \(T=f(t)\) that models the temperature \(T\) in degrees \(t\) hours since midnight.
- Predict when the temperature reaches 52 degrees.
33. The Bay of Fundy in eastern Canada is known for the highest tides in the world. The tides there rise and fall by as much as 50 ft. Assume the tidal cycle takes 11 hours.
- Find a formula for a function that models the height of the tides in the bay. Assume the low tide corresponds to h=0.
- Predict when the tide is over 40 feet.
34. An average person has a pulse rate of 72 beats per minute and a blood pressure of 120 over 80. Thus their heart is beating 72 times each minute and her blood pressure is oscillating from a low (diastolic) reading of 80 to a high (systolic) reading of 120 and down again after each beat.
a. Find the period of the person's blood pressure.
b. Sketch a graph that illustrates the blood pressure over 4 seconds.
c. Find a sine or cosine function that models her blood pressure after t seconds.
35. A Ferris wheel is 35 meters in diameter and boarded from a platform that is 3 meters above the ground. The six o’clock position on the Ferris wheel is level with the loading platform. The wheel completes 1 full revolution in 8 minutes. The function \(h(t)\) gives your height in meters above the ground t minutes after the wheel begins to turn.
a. Find the amplitude, midline, and period of \(h\left(t\right)\).
b. Find a formula for the height function \(h\left(t\right)\).
c. How high are you off the ground after 4 minutes?
36. A herd of elk oscillates from a high of 920 elk to a low of 570 during the year, hitting the lowest value in January (\(t = 0\)) due to food scarcity or abundance as well as the presence of predators among other factors.
a. Find a formula for the population, \(P\), in terms of the months since January, \(t\).
b. When does the elk herd have a population of 800 or more?
37. The sea ice area around the North Pole fluctuates between about 6 million square kilometers in September to 14 million square kilometers in March.
- Find a formula for the amount of sea ice around the North Pole over the course of the year.
- During which months are there less than 9 million square kilometers of sea ice?
- Answer
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1. Graph E
6. The period is /(P=\dfrac{2 \pi}{3} \) and the amplitude is \(A=4\).
14. \(y=2 \cos(2 \theta) \)
21. The period is /(P=\dfrac{2 \pi}{5}. The amplitude is \(A=4\). The centerline is at \(y = -2 \) . There is no phase shift.
27. \(y = 4 \sin (\dfrac{pi}{5}(x+1) \)
32. a. \(T = -7 \sin (\dfrac{pi}{12}t)+50 \) b. The temperature will reach 52 degrees when \(t \approx -1.115\) and \(t \approx 13.2 \) (just before 11 pm and shortly after 1 pm).