Exercises 6.3 Graphs of the Circular Functions

1.

1. Use your calculator to complete the table of values. Round values to hundredths.
   

   [latex]\theta[/latex]

   [latex]0[/latex]

   [latex]\dfrac{\pi}{12}[/latex]

   [latex]\dfrac{\pi}{6}[/latex]

   [latex]\dfrac{\pi}{4}[/latex]

   [latex]\dfrac{\pi}{3}[/latex]

   [latex]\dfrac{5\pi}{12}[/latex]

   [latex]\dfrac{\pi}{2}[/latex]

   [latex]\dfrac{7\pi}{12}[/latex]

   [latex]\dfrac{2\pi}{3}[/latex]

   [latex]\dfrac{3\pi}{4}[/latex]

   [latex]\dfrac{5\pi}{6}[/latex]

   [latex]\dfrac{11\pi}{12}[/latex]

   [latex]\pi[/latex]

   [latex]\cos \theta[/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]

2. Sketch a graph of [latex]~~y = \cos \theta~~[/latex] on the grid.
   

2.

1. Use your calculator to complete the table of values. Round values to hundredths.
   

   [latex]\theta[/latex]

   [latex]0[/latex]

   [latex]\dfrac{\pi}{12}[/latex]

   [latex]\dfrac{\pi}{6}[/latex]

   [latex]\dfrac{\pi}{4}[/latex]

   [latex]\dfrac{\pi}{3}[/latex]

   [latex]\dfrac{\pi}{2}[/latex]

   [latex]\dfrac{7\pi}{12}[/latex]

   [latex]\dfrac{2\pi}{3}[/latex]

   [latex]\dfrac{3\pi}{4}[/latex]

   [latex]\dfrac{5\pi}{6}[/latex]

   [latex]\dfrac{11\pi}{12}[/latex]

   [latex]\pi[/latex]

   [latex]\sin \theta[/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]

2. Sketch a graph of [latex]~~y = \sin \theta~~[/latex] on the grid.
   

3.

• Sketch a graph of [latex]y = \sin x[/latex] on each grid.
  
  

4.

• Sketch a graph of [latex]y = \cos x[/latex] on each grid.
  
  

5.

1. Sketch a graph of [latex]f(x) = \sin x[/latex] where [latex]x[/latex] is a real number.
2. State the domain and range of [latex]f(x) = \sin x{.}[/latex]

6.

1. Sketch a graph of [latex]g(x) = \cos x[/latex] where [latex]x[/latex] is a real number.
2. State the domain and range of [latex]g(x) = \cos x{.}[/latex]

7.

1. Sketch a graph of [latex]h(x) = \tan x[/latex] where [latex]x[/latex] is a real number.
2. State the domain and range of [latex]h(x) = \tan x{.}[/latex]

8.

• Sketch [latex]f(x) = \cos x[/latex] and [latex]g(x) = \sin x[/latex] on the same grid.

9.

1. Use the graph of [latex]y = \sin x[/latex] to estimate two solutions of the equation [latex]\sin x = 0.65{.}[/latex]
2. Use the unit circle to estimate two solutions of the equation [latex]\sin x = 0.35{.}[/latex]

10.

1. Use the graph of [latex]y = \sin x[/latex] to estimate two solutions of the equation [latex]\sin x = -0.2{.}[/latex]
2. Use the unit circle to estimate two solutions of the equation [latex]\sin x = -0.6{.}[/latex]

11.

1. Use the graph of [latex]y = \cos x[/latex] to estimate two solutions of the equation [latex]\cos x = -0.4{.}[/latex]
2. Use the unit circle to estimate two solutions of the equation [latex]\cos x = -0.8{.}[/latex]

12.

1. Use the graph of [latex]y = \cos x[/latex] to estimate two solutions of the equation [latex]\cos x = 0.15{.}[/latex]
2. Use the unit circle to estimate two solutions of the equation [latex]\cos x = 0.55{.}[/latex]

13.

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

14.

[latex]\tan x = 7[/latex]

15.

[latex]\tan x = -0.5[/latex]

16.

[latex]\tan x = -2.5[/latex]

17.

[latex]\sin x = 5.5\cos x[/latex]

18.

[latex]\sin x = -4.5\cos x[/latex]

19.

[latex]-2\sin x = 3\cos x[/latex]

20.

[latex]2\sin x = \cos x[/latex]

21.

[latex]5.8[/latex]

22.

[latex]2.9[/latex]

23.

[latex]3.7[/latex]

24.

[latex]5.1[/latex]

25.

[latex]1.8[/latex]

26.

[latex]4.4[/latex]

27.

[latex]\cos t = 0.74[/latex]

28.

[latex]\sin t = 0.58[/latex]

29.

[latex]\tan t = 1.6[/latex]

30.

[latex]\tan x = -0.6[/latex]

31.

[latex]\sin x = -0.72[/latex]

32.

[latex]\cos x = -0.48[/latex]

33.

[latex]\sin t = -1[/latex]

34.

[latex]\tan t = -1[/latex]

35.

[latex]\tan x = 1[/latex]

36.

[latex]\cos x = 0[/latex]

37.

[latex]\cos z = \dfrac{1}{2}[/latex]

38.

[latex]\sin z = \dfrac{1}{\sqrt{2}}[/latex]

39.

[latex]\tan s = -\sqrt{3}[/latex]

40.

[latex]\tan s = \dfrac{1}{\sqrt{3}}[/latex]

41.

[latex]\sin t = \dfrac{-1}{\sqrt{2}}[/latex]

42.

[latex]\cos t = \dfrac{-1}{2}[/latex]

43.

[latex]\cos x = \dfrac{-\sqrt{3}}{2}[/latex]

44.

[latex]\sin x = \dfrac{-\sqrt{3}}{2}[/latex]

45.

1. [latex]\displaystyle x = \sin 0.9[/latex]
2. [latex]\displaystyle \sin x = 0.9[/latex]

46.

1. [latex]\displaystyle x = \cos 0.73[/latex]
2. [latex]\displaystyle \cos x = 0.73[/latex]

47.

1. [latex]\displaystyle x = \tan 3.4[/latex]
2. [latex]\displaystyle \tan x = 3.4[/latex]

48.

1. [latex]\displaystyle x = \tan 5.8[/latex]
2. [latex]\displaystyle \tan x = 5.8[/latex]

49.

1. [latex]\displaystyle x = \cos 2.7[/latex]
2. [latex]\displaystyle \cos x = 2.7[/latex]

50.

1. [latex]\displaystyle x = \sin 1.2[/latex]
2. [latex]\sin x = 1.2[/latex]

51.

1. [latex]\displaystyle x = \sin \dfrac{\pi}{4}[/latex]
2. [latex]\displaystyle \sin x = \dfrac{\pi}{4}[/latex]

52.

1. [latex]\displaystyle x = \cos \dfrac{\pi}{6}[/latex]
2. [latex]\displaystyle \cos x = \dfrac{\pi}{6}[/latex]

53.

[latex]f(t) - 12\sin(2t - \dfrac{\pi}{4}),~~ t = \dfrac{3\pi}{4}[/latex]

54.

[latex]F(t) = 40\sin \left(\dfrac{t}{3} + \dfrac{\pi}{6}\right),~~ t = \pi[/latex]

55.

[latex]G(x) = 8\cos\left(\dfrac{x}{2} + \dfrac{2\pi}{3}\right),[/latex] [latex]~~ x = \dfrac{\pi}{3}[/latex]

56.

[latex]g(x) = -2\cos(3x - \pi),[/latex] [latex]~~ x = \dfrac{5\pi}{4}[/latex]

57.

[latex]H(\theta) = 6 - 2\tan \left(3\theta - \dfrac{\pi}{2}\right),[/latex] [latex]~~ \theta = \dfrac{5\pi}{6}[/latex]

58.

[latex]h(\theta) = 2 + 4\tan \left(\dfrac{\theta}{2} - \dfrac{3\pi}{2}\right),[/latex] [latex]~~ \theta = \dfrac{3\pi}{2}[/latex]

59.

When observed from earth, the moon looks like a disk that is partially visible and partially in shadow. The percentage of the disk that is visible can be approximated by

[latex]V(t) = 50\cos (0.21t) + 50[/latex]

where [latex]t[/latex] is the number of days since the last full moon.

1. Graph [latex]V(t)[/latex] in the window[latex][latex] \begin{aligned}[t] {X_{min}} = 0,~~ {X_{max}} = 30\\ {Y_{min}} = 0,~~ {Y_{max}} = 120\\ \end{aligned}[/latex]
2. Sketch the graph on the grid.
3. Label on your graph the points that correspond to full moon, half moon, and new moon. (New moon occurs when the part of the moon receiving sunlight is facing directly away from the earth.)
4. At what times during the lunar month is 25% of the moon visible? Mark those points on your graph.
5. During which days is less than 50% of the moon visible? Mark the corresponding points on your graph.

60.

The tide in Malibu is approximated by the function

[latex]h(t) = 2.5 - 2.5\cos(0.5t)[/latex]

measured in feet above low tide, where [latex]t[/latex] is the number of hours since the last low tide.

1. Graph [latex]h(t)[/latex] in the window[latex]{X_{min}} = 0,~~ {X_{max}} = 25\\ {Y_{min}} = 0,~~ {Y_{max}} = 6[/latex]
2. Sketch the graph on the grid.
3. Label on your graph the points that correspond to high tide and low tide.
4. How high is high tide, and at what times does it occur?
5. At what times during the 25-hour period is the tide 4 feet above low tide? Mark those points on your graph.
6. Kathie walks along the beach only when the tide is below 1 foot. Find the intervals on your graph when the tide is below 1 foot.

61.

The average daily high temperature in the town of Beardsley, Arizona, is approximated by the function

[latex]T(d) = 85.5 - 19.5\cos(0.0175d - 0.436)[/latex]

where the temperature is measured in degrees Fahrenheit, and [latex]d[/latex] is the number of days since January 1.

1. Graph [latex]T(d)[/latex] in the window[latex]{X_{min}} = 0,~~ {X_{max}} = 365\\ {Y_{min}} = 60,~~ {Y_{max}} = 110[/latex]
2. Sketch the graph on the grid.
3. Label on your graph the points that correspond to highest and lowest average temperature.
4. What is the hottest day and what is its average temperature? What is the day with the lowest average temperature, and what is that temperature?
5. At what times during the year are average high temperatures above [latex]90°{?}[/latex] Mark those points on your graph.

62.

A weight is suspended from the ceiling on a spring. The weight is pushed straight up, compressing the spring, then released. The height of the weight above the ground is given by the function

[latex]H(t) = 2 + 0.5\cos(6t)[/latex]

where the height is measured in meters, and [latex]t[/latex] is the number of seconds since the mass was released.

1. Graph [latex]H(t)[/latex] in the window[latex]{X_{min}} = 0,~~ {X_{max}} = 3\\ {Y_{min}} = 0,~~ {Y_{max}} = 3[/latex]
2. Sketch the graph on the grid.
3. Label on your graph the points that correspond to highest and lowest positions of the mass.
4. How high is highest point, and when is that height attained during the first 3 seconds?
5. How high is lowest point, and when is that height attained during the first 3 seconds?
6. Find the intervals during the first 3 seconds when the mass is less than 2 meters above the ground.

63.

64.

65.

66.

67.

Locate the four values [latex]t,~ 2\pi - t,~ \pi - t,~[/latex] and [latex]\pi + t[/latex] from Problem 63 on the graph of [latex]f(t) = \sin t{,}[/latex] on the graph of [latex]g(t) = \cos t{,}[/latex] and on the graph of [latex]h(t) = \tan t[/latex] shown below.

68.

Repeat Problem 67 for the values in Problem 64.

69.

Repeat Problem 67 for the values in Problem 65.

70.

Repeat Problem 67 for the values in Problem 66.

71.

[latex]f(x) = 9 - x^2[/latex]

72.

[latex]g(x) = 5 + \dfrac{1}{2}x^2[/latex]

73.

[latex]h(x) = \dfrac{-1}{x^2} + 2[/latex]

74.

[latex]j(x) = -3 + \dfrac{1}{x^2}[/latex]

75.

[latex]F(x) = \sqrt{x - 6}[/latex]

76.

[latex]G(x) = 6 - \sqrt{x}[/latex]

77.

[latex]H(x) = -\sqrt{4 - x^2}[/latex]

78.

[latex]J(x) = \sqrt{1 - x^2}[/latex]

79.

1. Sketch a graph of [latex]y = \cos x[/latex] using the guide points in the table below.
   

   [latex]x[/latex]	[latex]0[/latex]	[latex]\dfrac{\pi}{2}[/latex]	[latex]\pi[/latex]	[latex]\dfrac{3\pi}{2}[/latex]	[latex]2\pi[/latex]
   [latex]\cos x[/latex]	[latex]\hphantom{0000}[/latex]	[latex]\hphantom{0000}[/latex]	[latex]\hphantom{0000}[/latex]	[latex]\hphantom{0000}[/latex]	[latex]\hphantom{0000}[/latex]

2. State the domain and range of the function [latex]y = \cos x{.}[/latex]

80.

1. Sketch a graph of [latex]y = \tan x{,}[/latex] using the guide points in the table below.
   

   [latex]x[/latex]	[latex]0[/latex]	[latex]\dfrac{\pi}{4}[/latex]	[latex]\dfrac{\pi}{2}[/latex]	[latex]\dfrac{3\pi}{4}[/latex]	[latex]\pi[/latex]
   [latex]\tan x[/latex]	[latex]\hphantom{0000}[/latex]	[latex]\hphantom{0000}[/latex]	[latex]\hphantom{0000}[/latex]	[latex]\hphantom{0000}[/latex]	[latex]\hphantom{0000}[/latex]

2. State the domain and range of the function [latex]y = \tan x{.}[/latex]