1. Copyright © 2013, 2009, 2005 Pearson Education, Inc. 1 4 Graphs of the Circular Functions Copyright © 2013, 2009, 2005 Pearson Education, Inc. 1

2. 2 4.1 Graphs of the Sine and Cosine Functions 4.2 Translations of the Graphs of the Sine and Cosine Functions 4.3 Graphs of the Tangent and Cotangent Functions 4.4 Graphs of the Secant and Cosecant Functions 4.5Harmonic Motion Graphs of the Circular Functions 4

3. Copyright © 2013, 2009, 2005 Pearson Education, Inc. 3 Graphs of the Sine and Cosine Functions 4.1 Periodic Functions ▪ Graph of the Sine Function ▪ Graph of the Cosine Function ▪ Graphing Techniques, Amplitude, and Period ▪ Using a Trigonometric Model

4. Copyright © 2013, 2009, 2005 Pearson Education, Inc. 4 and compare to the graph of y = sin x. 4.1 Example 1 Graphing y = a sin x (page 137) The shape of the graph is the same as the shape of y = sin x. The range of is

5. Copyright © 2013, 2009, 2005 Pearson Education, Inc. 5 and compare to the graph of y = sin x. 4.1 Example 2 Graphing y = sin bx (page 138) The coefficient of x is, so b =, and the period is Divide the interval into four equal parts to get the x-values that will yield minimum and maximum points and x-intercepts. The endpoints are 0 and, and the three points between the endpoints are

6. Copyright © 2013, 2009, 2005 Pearson Education, Inc. 6 4.1 Example 2 Graphing y = sin bx (cont.) The x-values are

7. Copyright © 2013, 2009, 2005 Pearson Education, Inc. 7 4.1 Example 2 Graphing y = sin bx (cont.)

8. Copyright © 2013, 2009, 2005 Pearson Education, Inc. 8 4.1 Example 3 Graphing y = cos bx (page 139) The coefficient of x is, so b =, and the period is Divide the interval into four equal parts to get the x-values that will yield minimum and maximum points and x-intercepts. The x-values are 0, π, 2 π, 3 π, and 4 π.

9. Copyright © 2013, 2009, 2005 Pearson Education, Inc. 9 4.1 Example 3 Graphing y = cos bx (page 139)

10. Copyright © 2013, 2009, 2005 Pearson Education, Inc. 10 Graph y = –3 sin 2x. 4.1 Example 4 Graphing y = a sin bx (page 140) The coefficient of x is 2, so b = 2, and the period is The amplitude is |–3| = 3. Divide the interval into four equal parts to get the x-values that will yield minimum and maximum points and x-intercepts.

11. Copyright © 2013, 2009, 2005 Pearson Education, Inc. 11 4.1 Example 4 Graphing y = a sin bx (page 150)

12. Copyright © 2013, 2009, 2005 Pearson Education, Inc. 12 4.1 Example 5 Graphing y = a cos bx for b Equal to a Multiple of π (page 141) The amplitude is |2| = 2. Divide the interval [0, 4] into four equal parts to get the x-values that will yield minimum and maximum points and x-intercepts. The coefficient of x is, so b =, and the period is The x-values are 0, 1, 2, 3, and 4.

13. Copyright © 2013, 2009, 2005 Pearson Education, Inc. 13 4.1 Example 5 Graphing y = a cos bx for b Equal to a Multiple of π (cont.)

14. Copyright © 2013, 2009, 2005 Pearson Education, Inc. 14 4.1 Example 6 Determining an Equation for a Graph (page 141) Determine an equation of the form y = a cos bx or y = a sin bx, where b > 0 for the given graph. The graph is that of a sine function with period . The amplitude is The equation is

15. Copyright © 2013, 2009, 2005 Pearson Education, Inc. 15 4.1 Example 7(a) Interpreting a Sine Function Model (page 142) The average temperature (in °F) in Phoenix can be approximated by the function where x is the month and x = 1 corresponds to January, x = 2 corresponds to February, and so on. (a)To observe the graph over a two-year interval, graph f in the window [0, 25] by [0, 125]

16. Copyright © 2013, 2009, 2005 Pearson Education, Inc. 16 4.1 Example 7(b) Interpreting a Sine Function Model (cont.) b) According to this model, what is the average temperature during the month of October? October is month 10. Graph the function using a calculator, then find the value at x = 10. Alternatively, use a calculator to compute f(10).

17. Copyright © 2013, 2009, 2005 Pearson Education, Inc. 17 4.1 Example 7(c) Interpreting a Sine Function Model (page 142) (c)According to this model, what would be an approximation of the average yearly temperature in Phoenix? From the graph, it appears that the average yearly temperature is about 74°F.

18. Copyright © 2013, 2009, 2005 Pearson Education, Inc. 18 Translations of the Graphs of the Sine and Cosine Functions 4.2 Horizontal Translations ▪ Vertical Translations ▪ Combinations of Translations ▪ Determining a Trigonometric Model Using Curve Fitting

19. Copyright © 2013, 2009, 2005 Pearson Education, Inc. 19 4.2 Example 1 Graphing y = sin ( x – d ) (page 149) Step 2: Divide the period into four equal intervals: Step 1: b = 1, so find the interval whose length is

20. Copyright © 2013, 2009, 2005 Pearson Education, Inc. 20 4.2 Example 1 Graphing y = sin ( x – d ) (cont.) Step 3: Evaluate the function for each of the five x-values.

21. Copyright © 2013, 2009, 2005 Pearson Education, Inc. 21 4.2 Example 1 Graphing y = sin ( x – d ) (cont.) Steps 4 and 5: Plot the points found in the table and join them with a sinusoidal curve.

22. Copyright © 2013, 2009, 2005 Pearson Education, Inc. 22 4.2 Example 1 Graphing y = sin ( x – d ) (cont.) Note that this is the graph of y = sin x translated units to the left. Amplitude = 1 Period = Phase shift = units left No vertical translation

23. Copyright © 2013, 2009, 2005 Pearson Education, Inc. 23 4.2 Example 2 Graphing y = a cos ( x – d ) (page 150) Step 2: Divide the period into four equal intervals: Step 1: b = 1, so find the interval whose length is

24. Copyright © 2013, 2009, 2005 Pearson Education, Inc. 24 4.2 Example 2 Graphing y = a cos ( x – d ) (cont.) Step 3: Evaluate the function for each of the five x-values.

25. Copyright © 2013, 2009, 2005 Pearson Education, Inc. 25 4.2 Example 2 Graphing y = a cos ( x – d ) (cont.) Steps 4 and 5: Plot the points found in the table and join them with a sinusoidal curve.

26. Copyright © 2013, 2009, 2005 Pearson Education, Inc. 26 4.2 Example 2 Graphing y = a cos ( x – d ) (cont.) Note that this is the graph of y = –2 cos x translated units to the right. Amplitude = 2 Period =No vertical translation Phase shift = units right

27. Copyright © 2013, 2009, 2005 Pearson Education, Inc. 27 4.2 Example 3 Graphing y = a cos b ( x – d ) (page 150) Write the equation in the form y = a cos b(x – d). Step 1: b = 2, so find the interval whose length is

28. Copyright © 2013, 2009, 2005 Pearson Education, Inc. 28 4.2 Example 3 Graphing y = a cos b ( x – d ) (cont.) Step 2: Divide the period into four equal intervals: Step 3: Evaluate the function for each of the five x-values.

29. Copyright © 2013, 2009, 2005 Pearson Education, Inc. 29 4.2 Example 3 Graphing y = a cos b ( x – d ) (cont.) Steps 4 and 5: Plot the points found in the table and join them with a sinusoidal curve.

30. Copyright © 2013, 2009, 2005 Pearson Education, Inc. 30 4.2 Example 3 Graphing y = a cos b ( x – d ) (cont.) Amplitude, b = 2, so the period is Phase shift, units to the right as compared to the graph of

31. Copyright © 2013, 2009, 2005 Pearson Education, Inc. 31 4.2 Example 4 Graphing y = c + a cos bx (page 151) Step 1: Find the interval whose length is one period, Graph y = –2 + 3 cos 2x over two periods. Step 2: Divide the period into four equal intervals:

32. Copyright © 2013, 2009, 2005 Pearson Education, Inc. 32 4.2 Example 4 Graphing y = c + a cos bx (cont.) Step 3: Evaluate the function for each of the five x-values.

33. Copyright © 2013, 2009, 2005 Pearson Education, Inc. 33 4.2 Example 4 Graphing y = c + a cos bx (cont.) Steps 4 and 5: Plot the points found in the table and join them with a sinusoidal curve.

34. Copyright © 2013, 2009, 2005 Pearson Education, Inc. 34 4.2 Example 4 Graphing y = c + a cos bx (cont.) Note that this is the graph of y = 3 cos 2x translated 2 units down. Amplitude = 2 Phase shift : none Vertical translation: 2 units down Period =

35. Copyright © 2013, 2009, 2005 Pearson Education, Inc. 35 4.2 Example 5 Graphing y = c + a sin b ( x – d ) (page 152) Write the equation in the form y = c + a cos b(x – d). Step 1: Find the interval whose length is one period,

36. Copyright © 2013, 2009, 2005 Pearson Education, Inc. 36 4.2 Example 5 Graphing y = c + a sin b ( x – d ) (cont.) Step 2: Divide the period into four equal intervals: Step 3: Evaluate the function for each of the five x-values.

37. Copyright © 2013, 2009, 2005 Pearson Education, Inc. 37 4.2 Example 5 Graphing y = c + a sin b ( x – d ) (cont.)

38. Copyright © 2013, 2009, 2005 Pearson Education, Inc. 38 4.2 Example 5 Graphing y = c + a sin b ( x – d ) (cont.) Steps 4 and 5: Plot the points found in the table and join them with a sinusoidal curve.

39. Copyright © 2013, 2009, 2005 Pearson Education, Inc. 39 4.2 Example 5 Graphing y = c + a sin b ( x – d ) (cont.) Note that this is the graph of y = –2 sin 3x translated 4 units up and units to the right. Amplitude = 2 Vertical translation: 4 units up Period = Phase shift :

40. Copyright © 2013, 2009, 2005 Pearson Education, Inc. 40 4.2 Example 6(a) Modeling Temperature with a Sine Function (page 153) In Minneapolis, the average monthly temperatures include 13.0° in January, 47° in April, 73° in July, and 49° in October.* On average, January is the coldest month of the year, and July is the warmest. *Source: National Oceanic and Atmospheric Administration (NOAA). (a)Using only the maximum and minimum temperatures, determine a function of the form where a, b, c, and d are constants, that models the average monthly temperature in Minneapolis. Let x represent the month, with January corresponding to x = 1.

41. Copyright © 2013, 2009, 2005 Pearson Education, Inc. 41 4.2 Example 6(a) Modeling Temperature with a Sine Function (cont.) Use the maximum and minimum average monthly temperatures to determine the amplitude a: The average of the maximum and minimum temperatures gives c: Since temperatures repeat every 12 months,

42. Copyright © 2013, 2009, 2005 Pearson Education, Inc. 42 4.2 Example 7(a) Modeling Temperature with a Sine Function (cont.) To determine the phase shift, observe that the minimum temperature occurs in January. Thus, when x = 1, b(x – d) must equal n an integer, since the sine function is minimum at these values. Using solve for d:

43. Copyright © 2013, 2009, 2005 Pearson Education, Inc. 43 4.2 Example 7(a) Modeling Temperature with a Sine Function (cont.)

44. Copyright © 2013, 2009, 2005 Pearson Education, Inc. 44 4.2 Example 7(b) Modeling Temperature with a Sine Function (page 163) (b)Use the sine regression feature of a graphing calculator to determine a second model for these data. Two years’ data.

45. Copyright © 2013, 2009, 2005 Pearson Education, Inc. 45 4.2 Example 7(b) Modeling Temperature with a Sine Function (cont.)

46. Copyright © 2013, 2009, 2005 Pearson Education, Inc. 46 4.2 Example 7(c) Modeling Temperature with a Sine Function (page 163) (c)Use the models from parts (a) and (b) to predict the average temperature for September in Minneapolis. Compare the results. September is month 9. Using the model in part (a), we have Using the model in part (b), we have

47. Copyright © 2013, 2009, 2005 Pearson Education, Inc. 47 Graphs of the Tangent and Cotangent Functions 4.3 Graph of the Tangent Function ▪ Graph of the Cotangent Function ▪ Graphing Techniques

48. Copyright © 2013, 2009, 2005 Pearson Education, Inc. 48 4.3 Example 1 Graphing y = tan bx (page 162) Step 1: Find the period and locate the vertical asymptotes. Period = The asymptotes have the form

49. Copyright © 2013, 2009, 2005 Pearson Education, Inc. 49 4.3 Example 1 Graphing y = tan bx (cont.) Step 2: Sketch the two vertical asymptotes.

50. Copyright © 2013, 2009, 2005 Pearson Education, Inc. 50 4.3 Example 1 Graphing y = tan bx (cont.) Step 3: Divide the interval into four equal intervals. Step 4: Evaluate the function for the three middle x-values to find the first-quarter point, midpoint, and third-quarter point.

51. Copyright © 2013, 2009, 2005 Pearson Education, Inc. 51 4.3 Example 1 Graphing y = tan bx (cont.)

52. Copyright © 2013, 2009, 2005 Pearson Education, Inc. 52 4.3 Example 1 Graphing y = tan bx (cont.) Step 5: Join the points with a smooth curve, approaching the vertical asymptotes.

53. Copyright © 2013, 2009, 2005 Pearson Education, Inc. 53 4.3 Example 2 Graphing y = a tan bx (page 163) Step 1: Find the period and locate the vertical asymptotes. The asymptotes have the form

54. Copyright © 2013, 2009, 2005 Pearson Education, Inc. 54 4.3 Example 2 Graphing y = a tan bx (cont.) Step 2: Sketch the two vertical asymptotes.

55. Copyright © 2013, 2009, 2005 Pearson Education, Inc. 55 4.3 Example 2 Graphing y = a tan bx (cont.) Step 3: Divide the interval into four equal intervals. Step 4: Evaluate the function for the three middle x-values to find the first-quarter point, midpoint, and third-quarter point.

56. Copyright © 2013, 2009, 2005 Pearson Education, Inc. 56 4.3 Example 2 Graphing y = a tan bx (cont.)

57. Copyright © 2013, 2009, 2005 Pearson Education, Inc. 57 4.3 Example 2 Graphing y = a tan bx (cont.) Step 5: Join the points with a smooth curve, approaching the vertical asymptotes.

58. Copyright © 2013, 2009, 2005 Pearson Education, Inc. 58 4.3 Example 3 Graphing y = a cot bx (page 163) Step 1: Find the period and locate the vertical asymptotes. The asymptotes have the form bx = 0 and bx = π.

59. Copyright © 2013, 2009, 2005 Pearson Education, Inc. 59 4.3 Example 3 Graphing y = a cot bx (cont.) Step 2: Sketch the two vertical asymptotes.

60. Copyright © 2013, 2009, 2005 Pearson Education, Inc. 60 4.3 Example 3 Graphing y = a cot bx (cont.) Step 3: Divide the interval into four equal intervals. Step 4: Evaluate the function for the three middle x-values to find the first-quarter point, midpoint, and third-quarter point.

61. Copyright © 2013, 2009, 2005 Pearson Education, Inc. 61 4.3 Example 3 Graphing y = a cot bx (cont.)

62. Copyright © 2013, 2009, 2005 Pearson Education, Inc. 62 4.3 Example 3 Graphing y = a cot bx (cont.) Step 5: Join the points with a smooth curve, approaching the vertical asymptotes.

63. Copyright © 2013, 2009, 2005 Pearson Education, Inc. 63 4.3 Example 4 Graphing a Tangent Function With a Vertical Translation (page 164) Every value of this graph will be 3 units less than the corresponding value of y in y = tan x, causing the graph of y = –3 + tan x to be translated 3 units down compared with the graph of y = tan x.

64. Copyright © 2013, 2009, 2005 Pearson Education, Inc. 64 4.3 Example 5 Graphing a Cotangent Function With Vertical and Horizontal Translations (page 164) Step 1: Find the period and locate the vertical asymptotes. The asymptotes have the form bx = 0 and bx = π. The phase shift (horizontal translation) is, so the vertical asymptotes are

65. Copyright © 2013, 2009, 2005 Pearson Education, Inc. 65 4.3 Example 5 Graphing a Cotangent Function With Vertical and Horizontal Translations (cont.) Step 2: Sketch the two vertical asymptotes.

66. Copyright © 2013, 2009, 2005 Pearson Education, Inc. 66 4.3 Example 5 Graphing a Cotangent Function With Vertical and Horizontal Translations (cont.) Step 3: Divide the interval into four equal intervals. Step 4: Evaluate the function for the three middle x-values to find the first-quarter point, midpoint, and third-quarter point.

67. Copyright © 2013, 2009, 2005 Pearson Education, Inc. 67 4.3 Example 5 Graphing a Cotangent Function With Vertical and Horizontal Translations (cont.)

68. Copyright © 2013, 2009, 2005 Pearson Education, Inc. 68 4.3 Example 5 Graphing a Cotangent Function With Vertical and Horizontal Translations (cont.) Step 5: Join the points with a smooth curve, approaching the vertical asymptotes.

69. Copyright © 2013, 2009, 2005 Pearson Education, Inc. 69 4.3 Example 6a Determining an Equation for a Graph (page 165) Determine an equation for each graph. This is the graph of y = tan x, stretched vertically by a factor of 3. The equation is y = 3 tan x

70. Copyright © 2013, 2009, 2005 Pearson Education, Inc. 70 4.3 Example 6b Determining an Equation for a Graph (page 165) Determine an equation for each graph. This is the graph of y = cot x, the period is 2  instead of , and the graph is reflected across the y- axis. y = –cotbx with b > 0, b = ½. The equation is shifted up one unit.

71. Copyright © 2013, 2009, 2005 Pearson Education, Inc. 71 Graphs of the Secant and Cosecant Functions 4.4 Graph of the Secant Function ▪ Graph of the Cosecant Function ▪ Graphing Techniques ▪ Addition of Ordinates ▪ Connecting Graphs with Equations

72. Copyright © 2013, 2009, 2005 Pearson Education, Inc. 72 4.4 Example 1 Graphing y = a sec bx (page 172) Step 1: Graph the corresponding reciprocal function y = 3 cos 2x. Graph y = 3 sec 2x. One period is in the interval Dividing the interval into four equal parts gives the key points:

73. Copyright © 2013, 2009, 2005 Pearson Education, Inc. 73 4.4 Example 1 Graphing y = a sec bx (cont.)

74. Copyright © 2013, 2009, 2005 Pearson Education, Inc. 74 4.4 Example 1 Graphing y = a sec bx (cont.) Step 2: The vertical asymptotes of y = 3 sec 2x are at the x-intercepts of y = 3 cos 2x. Continuing this pattern to the left, there are also vertical asymptotes at

75. Copyright © 2013, 2009, 2005 Pearson Education, Inc. 75 4.4 Example 1 Graphing y = a sec bx (cont.) Step 3: Sketch the graph.

76. Copyright © 2013, 2009, 2005 Pearson Education, Inc. 76 4.4 Example 2 Graphing y = a csc ( x – d ) (page 172) Step 1: Graph the corresponding reciprocal function The phase shift is units left, so one period is in the interval

77. Copyright © 2013, 2009, 2005 Pearson Education, Inc. 77 4.4 Example 2 Graphing y = a csc ( x – d ) (cont.) Dividing the interval into four equal parts gives the key points:

78. Copyright © 2013, 2009, 2005 Pearson Education, Inc. 78 4.4 Example 2 Graphing y = a csc ( x – d ) (cont.) Step 2: The vertical asymptotes of are at the x-intercepts of Continuing this pattern to the left, there are also vertical asymptotes at

79. Copyright © 2013, 2009, 2005 Pearson Education, Inc. 79 4.4 Example 2 Graphing y = a csc ( x – d ) (cont.) Step 3: Sketch the graph.

80. Copyright © 2013, 2009, 2005 Pearson Education, Inc. 80 4.4 Example 3a Determining an Equation for a Graph (page 173) Determine an equation for the graph. Graph of y = sec x with period of 4 .

81. Copyright © 2013, 2009, 2005 Pearson Education, Inc. 81 4.4 Example 3b Determining an Equation for a Graph (page 173) Determine an equation for the graph. This is the graph of y = csc x translated one unit down. y = –1 + csc x

82. Copyright © 2013, 2009, 2005 Pearson Education, Inc. 82 Harmonic Motion 4.5 Simple Harmonic Motion ▪ Damped Oscillatory Motion

83. Copyright © 2013, 2009, 2005 Pearson Education, Inc. 83 4.5 Example 1 Modeling the Motion of a Spring (page 177) Suppose that an object is attached to a coiled spring. It is pulled down a distance of 16 cm from its equilibrium position and then released. The time for one complete oscillation is 6 seconds.

84. Copyright © 2013, 2009, 2005 Pearson Education, Inc. 84 4.5 Example 1(a) Modeling the Motion of a Spring (page 177) (a)Give an equation that models the position of the object at time t. When the object is released at t = 0, the object is at distance −16 cm from equilibrium. Since the time needed to complete one oscillation is 6 sec, P = 6, and

85. Copyright © 2013, 2009, 2005 Pearson Education, Inc. 85 4.5 Example 1(b, c) Modeling the Motion of a Spring (page 177) (b)Determine the position at t = 1.5 seconds. At t = 1.5 seconds, the object is at the equilibrium position. (c)Find the frequency. The frequency is the reciprocal of the period.

86. Copyright © 2013, 2009, 2005 Pearson Education, Inc. 86 4.5 Example 2 Analyzing Harmonic Motion (page 178) Suppose that an object oscillates according to the model s(t) = 2.5 sin 5t, where t is in seconds and s(t) is in meters. Analyze the motion. a = 2.5, so the object oscillates 2.5 meters in either direction from the starting point. and The motion is harmonic because the model is of the form