Chapter 7: Circular Functions

Exercises 7.1 Transformations of Graphs

Skills

  1. Identify the amplitude, period, and midline of a circular function #1–8, 23–30
  2. Graph a circular function #9–16, 31–44
  3. Find a formula for the graph of a circular function #17–30
  4. Model periodic phenomena with circular functions #45–52
  5. Graph transformations of the tangent function #53–58
  6. Solve trigonometric equations graphically #59–70

 

Suggested Homework Problems

Problems: #8, 14, 32, 38, 20, 24, 46, 56, 70

 

Exercises Homework 7.1

Exercise Group

For Problems 1–8, state the amplitude, period, and midline of the graph.

1.

[latex]y = -3+2\sin x[/latex]

2.

[latex]y = 4-3\cos x[/latex]

3.

[latex]y = -\cos 4x[/latex]

4.

[latex]y = -\sin 3x[/latex]

5.

[latex]y = -5\sin \dfrac{x}{3}[/latex]

6.

[latex]y = 6\cos \dfrac{x}{2}[/latex]

7.

[latex]y = 1-\cos \pi x[/latex]

8.

[latex]y = 2+\sin 2\pi x[/latex]

Exercise Group

In Problems 9–16, we use transformations to sketch graphs of the functions in Problems 1–8. Sketch one cycle of each graph by hand and label scales on the axes.

9.
  1. [latex]\displaystyle y = \sin x[/latex]
  2. [latex]\displaystyle y = 2\sin x[/latex]
  3. [latex]\displaystyle y = -3+2\sin x[/latex]
10.
  1. [latex]\displaystyle y = \cos x[/latex]
  2. [latex]\displaystyle y = -3\cos x[/latex]
  3. [latex]\displaystyle y = 4-3\cos x[/latex]
11.
  1. [latex]\displaystyle y = \cos x[/latex]
  2. [latex]\displaystyle y = \cos 4x[/latex]
  3. [latex]\displaystyle y = -\cos x[/latex]
12.
  1. [latex]\displaystyle y = \sin x[/latex]
  2. [latex]\displaystyle y = \sin 3x[/latex]
  3. [latex]\displaystyle y = -\sin 3x[/latex]
13.
  1. [latex]\displaystyle y = \sin x[/latex]
  2. [latex]\displaystyle y = \sin \dfrac{x}{3}[/latex]
  3. [latex]\displaystyle y = -5\sin \dfrac{x}{3}[/latex]
14.
  1. [latex]\displaystyle y = \cos x[/latex]
  2. [latex]\displaystyle y = \cos \dfrac{x}{2}[/latex]
  3. [latex]\displaystyle y = 6\cos \dfrac{x}{2}[/latex]
15.
  1. [latex]\displaystyle y = \cos x[/latex]
  2. [latex]\displaystyle y = \cos \pi x[/latex]
  3. [latex]\displaystyle y = 1-\cos \pi x[/latex]
16.
  1. [latex]\displaystyle y = \sin x[/latex]
  2. [latex]\displaystyle y = \sin 2\pi x[/latex]
  3. [latex]\displaystyle y = 2+\sin 2\pi x[/latex]

Exercise Group

For Problems 17–22, write an equation for the graph using sine or cosine.

17.

sinusoidal graph

18.

sinusoidal graph

19.

sinusoidal graph

20.

sinusoidal graph

21.

sinusoidal graph

22.

sinusoidal graph

Exercise Group

For Problems 23–30,

  1. State the amplitude, period, and midline of the graph.
  2. Write an equation for the graph using sine or cosine.
23.

sinusoidal graph

24.

sinusoidal graph

25.

sinusoidal graph

26.

sinusoidal graph

27.

sinusoidal graph

28.

sinusoidal graph

29.

sinusoidal graph

30.

sinusoidal graph

Exercise Group

In Problems 31–36, we use a table of values to sketch circular functions.

  1. Complete the table of values for the function.
  2. Sketch a graph of the function and label the scales on the axes.
31.

[latex]y=2-5\cos 2t[/latex]

[latex]t[/latex] [latex]2t[/latex] [latex]\cos 2t[/latex] [latex]-5\cos 2t[/latex] [latex]2-5\cos 2t[/latex]
[latex]\hphantom{0000}[/latex] [latex]0[/latex] [latex]\hphantom{0000}[/latex] [latex]\hphantom{0000}[/latex] [latex]\hphantom{0000}[/latex]
[latex]\hphantom{0000}[/latex] [latex]\dfrac{\pi}{2}[/latex] [latex]\hphantom{0000}[/latex] [latex]\hphantom{0000}[/latex] [latex]\hphantom{0000}[/latex]
[latex]\hphantom{0000}[/latex] [latex]\pi[/latex] [latex]\hphantom{0000}[/latex] [latex]\hphantom{0000}[/latex] [latex]\hphantom{0000}[/latex]
[latex]\hphantom{0000}[/latex] [latex]\dfrac{3\pi}{2}[/latex] [latex]\hphantom{0000}[/latex] [latex]\hphantom{0000}[/latex] [latex]\hphantom{0000}[/latex]
[latex]\hphantom{0000}[/latex] [latex]2\pi[/latex] [latex]\hphantom{0000}[/latex] [latex]\hphantom{0000}[/latex] [latex]\hphantom{0000}[/latex]

grid

32.

[latex]y=-2+4\sin 3t[/latex]

[latex]t[/latex] [latex]3t[/latex] [latex]\sin 3t[/latex] [latex]4\sin 3t[/latex] [latex]-2+4\sin 3t[/latex]
[latex]\hphantom{0000}[/latex] [latex]0[/latex] [latex]\hphantom{0000}[/latex] [latex]\hphantom{0000}[/latex] [latex]\hphantom{0000}[/latex]
[latex]\hphantom{0000}[/latex] [latex]\dfrac{\pi}{2}[/latex] [latex]\hphantom{0000}[/latex] [latex]\hphantom{0000}[/latex] [latex]\hphantom{0000}[/latex]
[latex]\hphantom{0000}[/latex] [latex]\pi[/latex] [latex]\hphantom{0000}[/latex] [latex]\hphantom{0000}[/latex] [latex]\hphantom{0000}[/latex]
[latex]\hphantom{0000}[/latex] [latex]\dfrac{3\pi}{2}[/latex] [latex]\hphantom{0000}[/latex] [latex]\hphantom{0000}[/latex] [latex]\hphantom{0000}[/latex]
[latex]\hphantom{0000}[/latex] [latex]2\pi[/latex] [latex]\hphantom{0000}[/latex] [latex]\hphantom{0000}[/latex] [latex]\hphantom{0000}[/latex]

graph

33.

[latex]y=1+3\cos \dfrac{t}{2}[/latex]

[latex]t[/latex] [latex]\dfrac{t}{2}[/latex] [latex]\cos \dfrac{t}{2}[/latex] [latex]3\cos \dfrac{t}{2}[/latex] [latex]1+3\cos \dfrac{t}{2}[/latex]
[latex]\hphantom{0000}[/latex] [latex]0[/latex] [latex]\hphantom{0000}[/latex] [latex]\hphantom{0000}[/latex] [latex]\hphantom{0000}[/latex]
[latex]\hphantom{0000}[/latex] [latex]\dfrac{\pi}{2}[/latex] [latex]\hphantom{0000}[/latex] [latex]\hphantom{0000}[/latex] [latex]\hphantom{0000}[/latex]
[latex]\hphantom{0000}[/latex] [latex]\pi[/latex] [latex]\hphantom{0000}[/latex] [latex]\hphantom{0000}[/latex] [latex]\hphantom{0000}[/latex]
[latex]\hphantom{0000}[/latex] [latex]\dfrac{3\pi}{2}[/latex] [latex]\hphantom{0000}[/latex] [latex]\hphantom{0000}[/latex] [latex]\hphantom{0000}[/latex]
[latex]\hphantom{0000}[/latex] [latex]2\pi[/latex] [latex]\hphantom{0000}[/latex] [latex]\hphantom{0000}[/latex] [latex]\hphantom{0000}[/latex]

graph

34.

[latex]y=-2-3\sin \dfrac{t}{4}[/latex]

[latex]t[/latex] [latex]\dfrac{t}{4}[/latex] [latex]\sin \dfrac{t}{4}[/latex] [latex]3\sin \dfrac{t}{4}[/latex] [latex]-2-3\sin \dfrac{t}{4}[/latex]
[latex]\hphantom{0000}[/latex] [latex]0[/latex] [latex]\hphantom{0000}[/latex] [latex]\hphantom{0000}[/latex] [latex]\hphantom{0000}[/latex]
[latex]\hphantom{0000}[/latex] [latex]\dfrac{\pi}{2}[/latex] [latex]\hphantom{0000}[/latex] [latex]\hphantom{0000}[/latex] [latex]\hphantom{0000}[/latex]
[latex]\hphantom{0000}[/latex] [latex]\pi[/latex] [latex]\hphantom{0000}[/latex] [latex]\hphantom{0000}[/latex] [latex]\hphantom{0000}[/latex]
[latex]\hphantom{0000}[/latex] [latex]\dfrac{3\pi}{2}[/latex] [latex]\hphantom{0000}[/latex] [latex]\hphantom{0000}[/latex] [latex]\hphantom{0000}[/latex]
[latex]\hphantom{0000}[/latex] [latex]2\pi[/latex] [latex]\hphantom{0000}[/latex] [latex]\hphantom{0000}[/latex] [latex]\hphantom{0000}[/latex]

graph

35.

[latex]y=-3+2\sin \dfrac{t}{3}[/latex]

[latex]\hphantom{0000}[/latex] [latex]\hphantom{0000}[/latex] [latex]\hphantom{0000}[/latex] [latex]\hphantom{0000}[/latex] [latex]\hphantom{0000}[/latex]
[latex]\hphantom{0000}[/latex] [latex]\hphantom{0000}[/latex] [latex]\hphantom{0000}[/latex] [latex]\hphantom{0000}[/latex] [latex]\hphantom{0000}[/latex]
[latex]\hphantom{0000}[/latex] [latex]\hphantom{0000}[/latex] [latex]\hphantom{0000}[/latex] [latex]\hphantom{0000}[/latex] [latex]\hphantom{0000}[/latex]
[latex]\hphantom{0000}[/latex] [latex]\hphantom{0000}[/latex] [latex]\hphantom{0000}[/latex] [latex]\hphantom{0000}[/latex] [latex]\hphantom{0000}[/latex]
[latex]\hphantom{0000}[/latex] [latex]\hphantom{0000}[/latex] [latex]\hphantom{0000}[/latex] [latex]\hphantom{0000}[/latex] [latex]\hphantom{0000}[/latex]
[latex]\hphantom{0000}[/latex] [latex]\hphantom{0000}[/latex] [latex]\hphantom{0000}[/latex] [latex]\hphantom{0000}[/latex] [latex]\hphantom{0000}[/latex]

graph

36.

[latex]y=-1+4\cos \dfrac{t}{6}[/latex]

[latex]\hphantom{0000}[/latex] [latex]\hphantom{0000}[/latex] [latex]\hphantom{0000}[/latex] [latex]\hphantom{0000}[/latex] [latex]\hphantom{0000}[/latex]
[latex]\hphantom{0000}[/latex] [latex]\hphantom{0000}[/latex] [latex]\hphantom{0000}[/latex] [latex]\hphantom{0000}[/latex] [latex]\hphantom{0000}[/latex]
[latex]\hphantom{0000}[/latex] [latex]\hphantom{0000}[/latex] [latex]\hphantom{0000}[/latex] [latex]\hphantom{0000}[/latex] [latex]\hphantom{0000}[/latex]
[latex]\hphantom{0000}[/latex] [latex]\hphantom{0000}[/latex] [latex]\hphantom{0000}[/latex] [latex]\hphantom{0000}[/latex] [latex]\hphantom{0000}[/latex]
[latex]\hphantom{0000}[/latex] [latex]\hphantom{0000}[/latex] [latex]\hphantom{0000}[/latex] [latex]\hphantom{0000}[/latex] [latex]\hphantom{0000}[/latex]
[latex]\hphantom{0000}[/latex] [latex]\hphantom{0000}[/latex] [latex]\hphantom{0000}[/latex] [latex]\hphantom{0000}[/latex] [latex]\hphantom{0000}[/latex]

graph

Exercise Group

For Problems 37–44, label the scales on the axes for the graph.

37.

[latex]y=3-4\sin 2x[/latex]
sinusoidal graph, no scale on axes

38.

[latex]y=2\cos 5x+2[/latex]
sinusoidal graph

39.

[latex]y=\dfrac{1}{2} \sin 3x+\dfrac{3}{2}[/latex]
sinusoidal graph, no scale

40.

[latex]y = \dfrac{2}{5}\cos 6x +\dfrac{4}{5}[/latex]
sinusoidal graph

41.

[latex]50-30 \sin \dfrac{x}{4}[/latex]
sinusoidal graph, no scale

42.

[latex]25 \cos \dfrac{x}{3} + 15[/latex]
sinusoidal graph

43.

[latex]y = 4 \sin \pi x - 3[/latex]
sinusoidal graph

44.

[latex]y = \dfrac{1}{2} \cos \dfrac{\pi x}{2} + 2[/latex]
sinusoidal graph

45.

The height of the tide in Cabot Cove can be approximated by a sinusoidal function. At 5 am on July 23, the water level reached its high mark at the 20-foot line on the pier, and at 11 am, the water level was at its lowest at the 4-foot line.

  1. Sketch a graph of [latex]W(t){,}[/latex] the water level as a function of time, from 5 am on July 23 to 5 am on July 24.
  2. Write an equation for the function.

46.

The population of mosquitoes at Marsh Lake is a sinusoidal function of time. The population peaks around June 1, at about 6000 mosquitoes per square kilometer, and is smallest on December 1, at 1000 mosquitoes per square kilometer.

  1. Sketch a graph of [latex]M(t){,}[/latex] the number of mosquitoes as a function of the month, where [latex]t=0[/latex] on June 1.
  2. Write an equation for the function.

47.

The paddlewheel on the Delta Queen steamboat is 28 feet in diameter and is rotating once every ten seconds. The bottom of the paddlewheel is 4 feet below the surface of the water.

  1. The ship’s logo is painted on one of the paddlewheel blades. At [latex]t=0{,}[/latex] the blade with the logo is at the top of the wheel. Sketch a graph of the logo’s height above the water as a function of [latex]t{.}[/latex]
  2. Write an equation for the function.

48.

Delbert’s bicycle wheel is 24 inches in diameter, and he has a light attached to the spokes 10 inches from the center of the wheel. It is dark, and he is cycling home slowly from work. The bicycle wheel makes one revolution every second.

  1. At [latex]t=0{,}[/latex] the light is at its highest point the bicycle wheel. Sketch a graph of the light’s height as a function of [latex]t{.}[/latex]
  2. Write an equation for the function.

Exercise Group

For Problems 49–52, write an equation for the sinusoidal function whose graph is shown.

49.

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. Time is measured in months, starting at the winter solstice.
sinusoidal graph

50.

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. The weight oscillates between its minimum and maximum height.
sinusoidal graph

51.

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

52.

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) and 100 (at full moon) over a 28-day cycle.
sinusoidal graph

Exercise Group

For Problems 53–58,

  1. Make a table of values and sketch a graph of the function.
  2. Give its period and midline.
53.

[latex]y=\tan 2x[/latex]

54.

[latex]y=\tan 4x[/latex]

55.

[latex]y=4+2\tan 3x[/latex]

56.

[latex]y=3+\dfrac{1}{2} \tan 2x[/latex]

57.

[latex]y=3-\tan \dfrac{x}{4}[/latex]

58.

[latex]y=1-2\tan \dfrac{x}{3}[/latex]

Exercise Group

For Problems 59–64, use the graph to find all solutions between [latex]0[/latex] and [latex]2\pi{.}[/latex]

59.

[latex]3\cos 4x = 1.5[/latex]
sinusoidal graph and horizontal line

60.

[latex]2\sin 3x = -\sqrt{2}[/latex]
sinusoidal graph and horizontal line

61.

[latex]2+3\sin 2x = 0.5[/latex]
sinusoidal graph and horizontal line

62.

[latex]2+4\cos 2x = 4[/latex]
sinusoidal graph and horizontal line

63.

[latex]-3+\tan 3x = -2[/latex]
transformed tangent and horizontal line

64.

[latex]2+\tan 4x = 3[/latex]
transformed tangent and horizontal line

Exercise Group

For Problems 65–70,

  1. Use a calculator to graph the function for [latex]0\le x \le 2\pi{.}[/latex]
  2. Use the intersect feature to find all solutions between [latex]0[/latex] and [latex]2\pi{.}[/latex] Round your answers to hundredths.
65.
  1. [latex]\displaystyle f(x)=3\sin 2x[/latex]
  2. [latex]\displaystyle 3\sin 2x = -1.5[/latex]
66.
  1. [latex]\displaystyle g(x)=-2\cos 3x[/latex]
  2. [latex]\displaystyle -2\cos 3x = 1[/latex]
67.
  1. [latex]\displaystyle h(x)=2 - 4\cos \dfrac{x}{4}[/latex]
  2. [latex]\displaystyle 2 - 4\cos \dfrac{x}{4} = 0[/latex]
68.
  1. [latex]\displaystyle H(x)= 3+2\sin \dfrac{x}{2}[/latex]
  2. [latex]\displaystyle 3+2\sin \dfrac{x}{2} = 5[/latex]
69.
  1. [latex]\displaystyle G(x)= -1 + 3 \cos 3x[/latex]
  2. [latex]\displaystyle -1 + 3 \cos 3x = 1[/latex]
70.
  1. [latex]\displaystyle F(x)= 4 - 3\sin 2x[/latex]
  2. [latex]\displaystyle 4 - 3\sin 2x = 2.5[/latex]

License

Trigonometry Copyright © 2024 by Bimal Kunwor; Donna Densmore; Jared Eusea; and Yi Zhen. All Rights Reserved.

Share This Book