Problem Set 48: Graphs of the Sine and Cosine Function

1. Why are the sine and cosine functions called periodic functions?

2. How does the graph of [latex]y=\sin x[/latex] compare with the graph of [latex]y=\cos x[/latex]? Explain how you could horizontally translate the graph of [latex]y=\sin x[/latex] to obtain [latex]y=\cos x[/latex].

3. For the equation [latex]A\cos(Bx+C)+D[/latex], what constants affect the range of the function and how do they affect the range?

4. How does the range of a translated sine function relate to the equation [latex]y=A\sin(Bx+C)+D[/latex]?

5. How can the unit circle be used to construct the graph of [latex]f(t)=\sin t[/latex]?

6. [latex]f(x)=2\sin x[/latex]

7. [latex]f(x)=\frac{2}{3}\cos x[/latex]

8. [latex]f(x)=−3\sin x[/latex]

9. [latex]f(x)=4\sin x[/latex]

10. [latex]f(x)=2\cos x[/latex]

11. [latex]f(x)=\cos(2x)[/latex]

12. [latex]f(x)=2\sin\left(\frac{1}{2}x\right)[/latex]

13. [latex]f(x)=4\cos(\pi x)[/latex]

14. [latex]f(x)=3\cos\left(\frac{6}{5}x\right)[/latex]

15. [latex]y=3\sin(8(x+4))+5[/latex]

16. [latex]y=2\sin(3x−21)+4[/latex]

17. [latex]y=5\sin(5x+20)−2[/latex]

For the following exercises, graph one full period of each function, starting at [latex]x=0[/latex]. 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 [latex]x>0[/latex]. State the phase shift and vertical translation, if applicable. Round answers to two decimal places if necessary.

18. [latex]f(t)=2\sin\left(t−\frac{5\pi}{6}\right)[/latex]

19. [latex]f(t)=−\cos\left(t+\frac{\pi}{3}\right)+1[/latex]

20. [latex]f(t)=4\cos\left(2\left(t+\frac{\pi}{4}\right)\right)−3[/latex]

21. [latex]f(t)=−\sin\left(12t+\frac{5\pi}{3}\right)[/latex]

22. [latex]f(x)=4\sin\left(\frac{\pi}{2}(x−3)\right)+7[/latex]

23. Determine the amplitude, midline, period, and an equation involving the sine function for the graph shown in Figure 26.

A sinusoidal graph with amplitude of 2, range of [-5, -1], period of 4, and midline at y=-3.

Figure 26

24. Determine the amplitude, period, midline, and an equation involving cosine for the graph shown in Figure 27.

A graph with a cosine parent function, with amplitude of 3, period of pi, midline at y=-1, and range of [-4,2]

Figure 27

25. Determine the amplitude, period, midline, and an equation involving cosine for the graph shown in Figure 28.

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].

Figure 28

26. Determine the amplitude, period, midline, and an equation involving sine for the graph shown in Figure 29.

A sinusoidal graph with amplitude of 4, period of 10, midline at y=0, and range [-4,4].

Figure 29

27. Determine the amplitude, period, midline, and an equation involving cosine for the graph shown in Figure 30.

A graph with cosine parent function, range of function is [-4,4], amplitude of 4, period of 2.

Figure 30

28. Determine the amplitude, period, midline, and an equation involving sine for the graph shown in Figure 31.

A graph with sine parent function. Amplitude 2, period 2, midline y=0

Figure 31

29. Determine the amplitude, period, midline, and an equation involving cosine for the graph shown in Figure 32.

A graph with cosine parent function. Amplitude 2, period 2, midline y=1

Figure 32

30. Determine the amplitude, period, midline, and an equation involving sine for the graph shown in Figure 33.

A graph with a sine parent function. Amplitude 1, period 4 and midline y=0.

Figure 33

For the following exercises, let [latex]f(x)=\sin x[/latex].

31. On [0,2π), solve [latex]f(x)=\frac{1}{2}[/latex].

32. Evaluate [latex]f\left(\frac{\pi}{2}\right)[/latex].

33. On [0,2π), [latex]f(x)=\frac{\sqrt{2}}{2}[/latex]. Find all values of x.

34. On [0,2π), the maximum value(s) of the function occur(s) at what x-value(s)?

35. On [0,2π), the minimum value(s) of the function occur(s) at what x-value(s)?

36. Show that [latex]f(−x)=−f(x)[/latex].This means that [latex]f(x)=\sin x[/latex] is an odd function and possesses symmetry with respect to ________________.

For the following exercises, let [latex]f(x)=\cos x[/latex].

37. On [0,2π), solve the equation [latex]f(x)=\cos x=0[/latex].

38. On[0,2π), solve [latex]f(x)=\frac{1}{2}[/latex].

39. On [0,2π), find the x-intercepts of [latex]f(x)=\cos x[/latex].

40. On [0,2π), find the x-values at which the function has a maximum or minimum value.

41. On [0,2π), solve the equation [latex]f(x)=\frac{\sqrt{3}}{2}[/latex].

42. Graph [latex]h(x)=x+\sin x \text{ on}[0,2\pi][/latex]. Explain why the graph appears as it does.

43. Graph [latex]h(x)=x+\sin x[/latex] on[−100,100]. Did the graph appear as predicted in the previous exercise?

44. Graph [latex]f(x)=x\sin x[/latex] on [0,2π] and verbalize how the graph varies from the graph of [latex]f(x)=\sin x[/latex].

45. Graph [latex]f(x)=x\sin x[/latex] on the window [−10,10] and explain what the graph shows.

46. Graph [latex]f(x)=\frac{\sin x}{x}[/latex] on the window [−5π,5π] and explain what the graph shows.

47. 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.
a. Find the amplitude, midline, and period of h(t).
b. Find a formula for the height function h(t).
c. How high off the ground is a person after 5 minutes?