Section Exercises
Verbal
1. Why are the sine and cosine functions called periodic functions?
Show Solution
The sine and cosine functions have the property that f(x+P)=f(x) for a certain P. This means that the function values repeat for every P units on the x-axis.
2. How does the graph of y=sin x compare with the graph of y=cos x? Explain how you could horizontally translate the graph of y=sin x to obtain y=cos x.
3. For the equation A cos(Bx+C)+D,what constants affect the range of the function and how do they affect the range?
Show Solution
The absolute value of the constant A (amplitude) increases the total range and the constant D (vertical shift) shifts the graph vertically.
4. How does the range of a translated sine function relate to the equation y=A sin(Bx+C)+D?
5. How can the unit circle be used to construct the graph of f(t)=sin t?
Show Solution
At the point where the terminal side of t intersects the unit circle, you can determine that the sin t equals the y-coordinate of the point.
Graphical
For the following exercises, graph two full periods of each function and state the amplitude, period, and midline. State the maximum and minimum y-values and their corresponding x-values on one period for x>0. Round answers to two decimal places if necessary.
7. f(x)=2sin x
8. f(x)=23cos x
Show Solution
![A graph of (2/3)cos(x). Graph has amplitude of 2/3, period of 2pi, and range of [-2/3, 2/3].](https://s3-us-west-2.amazonaws.com/courses-images/wp-content/uploads/sites/3896/2019/03/07133050/CNX_Precalc_Figure_06_01_202.jpg)
amplitude: 23; period: 2π; midline: y=0; maximum: y=23 occurs at x=0; minimum: y=−23 occurs at x=π; for one period, the graph starts at 0 and ends at 2π
9. f(x)=−3sin x
10. f(x)=4sin x
Show Solution
![A graph of 4sin(x). Graph has amplitude of 4, period of 2pi, and range of [-4, 4].](https://s3-us-west-2.amazonaws.com/courses-images/wp-content/uploads/sites/3896/2019/03/07133053/CNX_Precalc_Figure_06_01_204.jpg)
amplitude: 4; period: 2π; midline: y=0; maximum y=4 occurs at x=π2; minimum: y=−4 occurs at x=3π2; one full period occurs from x=0 to x=2π
11. f(x)=2cos x
12. f(x)=cos(2x)
Show Solution
![A graph of cos(2x). Graph has amplitude of 1, period of pi, and range of [-1,1].](https://s3-us-west-2.amazonaws.com/courses-images/wp-content/uploads/sites/3896/2019/03/07133055/CNX_Precalc_Figure_06_01_206.jpg)
amplitude: 1; period: π; midline: y=0; maximum: y=1 occurs at x=π; minimum: y=−1 occurs at x=π2; one full period is graphed from x=0 to x=π
13. f(x)=2 sin(12x)
14. f(x)=4 cos(πx)
Show Solution
![A graph of 4cos(pi*x). Grpah has amplitude of 4, period of 2, and range of [-4, 4].](https://s3-us-west-2.amazonaws.com/courses-images/wp-content/uploads/sites/3896/2019/03/07133057/CNX_Precalc_Figure_06_01_208.jpg)
amplitude: 4; period: 2; midline: y=0; maximum: y=4 occurs at x=0; minimum: y=−4 occurs at x=1
15. f(x)=3 cos(65x)
16. y=3 sin(8(x+4))+5
Show Solution
![A graph of 3sin(8(x+4))+5. Graph has amplitude of 3, range of [2, 8], and period of pi/4.](https://s3-us-west-2.amazonaws.com/courses-images/wp-content/uploads/sites/3896/2019/03/07133059/CNX_Precalc_Figure_06_01_210.jpg)
amplitude: 3; period: π4; midline: y=5; maximum: y=8 occurs at x=0.12; minimum: y=2 occurs at x=0.516; horizontal shift: −4; vertical translation 5; one period occurs from x=0 to x=π4
17. y=2 sin(3x−21)+4
18. y=5 sin(5x+20)−2
Show Solution
![A graph of 5sin(5x+20)-2. Graph has an amplitude of 5, period of 2pi/5, and range of [-7,3].](https://s3-us-west-2.amazonaws.com/courses-images/wp-content/uploads/sites/3896/2019/03/07133102/CNX_Precalc_Figure_06_01_212.jpg)
amplitude: 5; period: 2π5; midline: y=−2; maximum: y=3 occurs at x=0.08; minimum: y=−7 occurs at x=0.71; phase shift: −4; vertical translation: −2; one full period can be graphed on x=0 to x=2π5
For the following exercises, graph one full period of each function, starting at x=0. For each function, state the amplitude, period, and midline. State the maximum and minimum y-values and their corresponding x-values on one period for x>0. State the phase shift and vertical translation, if applicable. Round answers to two decimal places if necessary.
19. f(t)=2sin(t−5π6)
20. f(t)=−cos(t+π3)+1
Show Solution
![A graph of -cos(t+pi/3)+1. Graph has amplitude of 1, period of 2pi, and range of [0,2]. Phase shifted pi/3 to the left.](https://s3-us-west-2.amazonaws.com/courses-images/wp-content/uploads/sites/3896/2019/03/07133104/CNX_Precalc_Figure_06_01_214.jpg)
amplitude: 1 ; period: 2π; midline: y=1; maximum: y=2 occurs at x=2.09; maximum: y=2 occurs at t=2.09; minimum: y=0 occurs at t=5.24; phase shift: −π3; vertical translation: 1; one full period is from t=0 to t=2π
21. f(t)=4cos(2(t+π4))−3
22. f(t)=−sin(12t+5π3)
Show Solution
![A graph of -sin((1/2)*t + 5pi/3). Graph has amplitude of 1, range of [-1,1], period of 4pi, and a phase shift of -10pi/3.](https://s3-us-west-2.amazonaws.com/courses-images/wp-content/uploads/sites/3896/2019/03/07133107/CNX_Precalc_Figure_06_01_216.jpg)
amplitude: 1; period: 4π; midline: y=0; maximum: y=1 occurs at t=11.52; minimum: y=−1 occurs at t=5.24; phase shift: −10π3; vertical shift: 0
23. f(x)=4sin(π2(x−3))+7
Determine the amplitude, midline, period, and an equation involving the sine function for the graph shown in (Figure).
Show Solution
amplitude: 2; midline: y=−3; period: 4; equation: f(x)=2sin(π2x)−3
24. Determine the amplitude, period, midline, and an equation involving cosine for the graph shown in (Figure).
25. Determine the amplitude, period, midline, and an equation involving cosine for the graph shown in (Figure).
Show Solution
amplitude: 2; period: 5; midline: y=3; equation: f(x)=−2cos(2π5x)+3
26. Determine the amplitude, period, midline, and an equation involving sine for the graph shown in (Figure).
27. Determine the amplitude, period, midline, and an equation involving cosine for the graph shown in (Figure).
Show Solution
amplitude: 4; period: 2; midline: y=0; equation: f(x)=−4cos(π(x−π2))
28. Determine the amplitude, period, midline, and an equation involving sine for the graph shown in (Figure).
29. Determine the amplitude, period, midline, and an equation involving cosine for the graph shown in (Figure).
Show Solution
amplitude: 2; period: 2; midline y=1; equation: f(x)=2cos(πx)+1
30. Determine the amplitude, period, midline, and an equation involving sine for the graph shown in (Figure).
Algebraic
For the following exercises, let f(x)=sin x.
31. On [0,2π),solve f(x)=0.
32. On [0,2π),solve f(x)=12.
33.Evaluate f(π2).
34. On [0,2π),f(x)=√22. Find all values of x.
35. On [0,2π),the maximum value(s) of the function occur(s) at what x-value(s)?
36. On [0,2π),the minimum value(s) of the function occur(s) at what x-value(s)?
37. Show that f(−x)=−f(x). This means that f(x)=sin x is an odd function and possesses symmetry with respect to ________________.
For the following exercises, let f(x)=cos x.
38. On [0,2π),solve the equation f(x)=cos x=0.
39. On [0,2π),solve f(x)=12.
40. On [0,2π),find the x-intercepts of f(x)=cos x.
41. On [0,2π),find the x-values at which the function has a maximum or minimum value.
42. On [0,2π),solve the equation f(x)=√32.
Technology
43. Graph h(x)=x+sin x on [0,2π]. Explain why the graph appears as it does.
44. Graph h(x)=x+sin x on [−100,100]. Did the graph appear as predicted in the previous exercise?
Show Solution
The graph appears linear. The linear functions dominate the shape of the graph for large values of x.
45. Graph f(x)=x sin x on [0,2π] and verbalize how the graph varies from the graph of f(x)=sin x.
46. Graph f(x)=x sin x on the window [−10,10] and explain what the graph shows.
Show Solution
The graph is symmetric with respect to the y-axis and there is no amplitude because the function is not periodic.
47. Graph f(x)=sin xx on the window [−5π,5π] and explain what the graph shows.
Real-World Applications
48. A Ferris wheel is 25 meters in diameter and boarded from a platform that is 1 meter 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 10 minutes. The function h(t) gives a person’s height in meters above the ground t minutes after the wheel begins to turn.
- Find the amplitude, midline, and period of h(t).
- Find a formula for the height function h(t).
- How high off the ground is a person after 5 minutes?
Show Solution
- Amplitude: 12.5; period: 10; midline: y=13.5;
- h(t)=12.5sin(π5(t−2.5))+13.5;
- 26 ft
![A sinusoidal graph with amplitude of 2, range of [-5, -1], period of 4, and midline at y=-3.](https://s3-us-west-2.amazonaws.com/courses-images/wp-content/uploads/sites/3896/2019/03/07133110/CNX_Precalc_Figure_06_01_218.jpg)
![A graph with a cosine parent function, with amplitude of 3, period of pi, midline at y=-1, and range of [-4,2]](https://s3-us-west-2.amazonaws.com/courses-images/wp-content/uploads/sites/3896/2019/03/07133116/CNX_Precalc_Figure_06_01_219.jpg)
![A graph with a cosine parent function with an amplitude of 2, period of 5, midline at y=3, and a range of [1,5].](https://s3-us-west-2.amazonaws.com/courses-images/wp-content/uploads/sites/3896/2019/03/07133119/CNX_Precalc_Figure_06_01_220.jpg)
![A sinusoidal graph with amplitude of 4, period of 10, midline at y=0, and range [-4,4].](https://s3-us-west-2.amazonaws.com/courses-images/wp-content/uploads/sites/3896/2019/03/07133122/CNX_Precalc_Figure_06_01_221.jpg)
![A graph with cosine parent function, range of function is [-4,4], amplitude of 4, period of 2.](https://s3-us-west-2.amazonaws.com/courses-images/wp-content/uploads/sites/3896/2019/03/07133125/CNX_Precalc_Figure_06_01_222.jpg)
