1. Copyright © 2014, 2010, 2007 Pearson Education, Inc.
Chapter 5
Trigonometric Functions
5.5 Part 1 Graphs of Sine and Cosine Functions
Copyright © 2014, 2010, 2007 Pearson Education, Inc. 1

2. Objectives:
Understand the graph of y = sin x.
Understand the graph of y = cos x.
Graph variations of y = sin x.
Graph variations of y = cos x.
Use vertical shifts of sine and cosine curves.
Model periodic behavior.

3. Basic Sine Curve © 2010 Pearson Education, Inc. All rights reserved
In this section, you will study techniques for sketching the graphs of the sine and cosine functions. The graph of the sine function is a sine curve.
Plotting y = sin x for common values of x and connecting the points with a smooth curve yields the following:

4. Sine Curve from the Unit Circle
If (x, y) is a point on the unit circle that is t units along the circumference from the point (1, 0), then (x, y) = (cos t, sin t). Therefore, the y values in the equality y = sin x are the y-coordinates of the points that are t units away from (1, 0) on the unit circle.
(t, y)
(x, y)

5. The Graph of y = sinx

6. Basic Sine Curve
In Figure below, the black portion of the graph represents one period of the function and is called one cycle of the sine curve. The gray portion of the graph indicates that the basic sine curve repeats indefinitely to the left and right.

7. Basic Sine Curve
We know that the domain of the sine function is the set of all real numbers. Moreover, the range of the function is the interval [–1, 1], and has a period of 2. Do you see how this information is consistent with the basic graph of the sine curve?

8. Basic Sine Curve
Since the period of the sine function is 2π, all possible values of the sine function will occur in any 2π interval, and in each adjoining 2π interval the exact pattern of values will be repeated.
Any 2π interval of values of the sine function is called “one period” of the sine function.
We will define the “primary period” of the sine function to be those values in the interval [0, 2π).

9. Basic Sine Curve
Note that the sine curve is symmetric with respect to the origin. The properties of symmetry follow from the fact that the sine function is an odd function.

10. Properties of the Basic Sine Curve (y = sin x)
The domain is
The range is [–1, 1].
The function is an odd function:
The period is

11. Sketching the Primary Period of the Sine Curve
To sketch the graph of the basic sine function by hand, it helps to note five key points in one period of the graph: the intercepts, maximum points, and minimum points (see below).

12. Sketching the Primary Period of the Sine Curve
In a rectangular coordinate system, mark the positions of 0 and 2π on the x-axis.
Divide this interval (the period 2π) on the x-axis into four intervals (π/2, π, and 3π/2) and label the endpoints of each (quarter points).
For each of these five numbers, the corresponding sine values are 0, 1, 0 -1, 0.
Plot the (x, sin x) pairs and connect them with a smooth curve.
Graph may be extended left and right.

13. Figure: y = sin x, 0 < x < 2π

14. y = sin x, -∞ < x < ∞

15. Summary Basic Sine Curve

16. Example:
True or False: The graph of y = sin x is symmetric with respect to the y – axis. Explain. False, the graph of y = sin x is symmetric with respect to the origin and an odd function.

17. Your Turn:
True or False: The domain of the graph of y = sin x is [-1, 1]. Explain.

18. The graph of the cosine function is a cosine curve.
© 2010 Pearson Education, Inc. All rights reserved
Basic Cosine Curve
The graph of the cosine function is a cosine curve.
Plotting y = cos x for common values of x and connecting the points with a smooth curve yields the following:

19. Cosine Curve from the Unit Circle
© 2010 Pearson Education, Inc. All rights reserved
Cosine Curve from the Unit Circle
If (x, y) is a point on the unit circle that is t units along the circumference from the point (1, 0), then (x, y) = (cos t, sin t). Therefore, the y values in the equality y = cos x are the x-coordinates of the points that are t units away from (1, 0) on the unit circle.

20. The Graph of y = cosx

21. Basic Cosine Curve
In Figure below, the black portion of the graph represents one period of the function and is called one cycle of the cosine curve. The gray portion of the graph indicates that the basic cosine curve repeats indefinitely to the left and right.

22. Basic Cosine Curve
We know that the domain of the cosine function is the set of all real numbers. Moreover, the range of the function is the interval [–1, 1], and has a period of 2. Do you see how this information is consistent with the basic graph of the cosine curve?

23. Basic Cosine Curve
Since the period of the cosine function is 2π, all possible values of the cosine function will occur in any 2π interval, and in each adjoining 2π interval the exact pattern of values will be repeated.
Any 2π interval of values of the cosine function is called “one period” of the cosine function.
We will define the “primary period” of the cosine function to be those values in the interval [0, 2π).

24. Basic Cosine Curve
Note that the cosine curve is symmetric with respect to the y-axis. These properties of symmetry follow from the fact that the cosine function is an even function.

25. Properties of the Basic Cosine Curve (y = cos x)
The domain is
The range is [–1, 1].
The function is an even function:
The period is

26. Sketching the Primary Period of the Cosine Curve
To sketch the graphs of the basic cosine function by hand, it helps to note five key points in one period of each graph: the intercepts, maximum points, and minimum points (see below).

27. Sketching the Primary Period of the Cosine Curve
In a rectangular coordinate system, mark the positions of 0 and 2π on the x-axis.
Divide this interval (the period 2π) on the x-axis into four intervals (π/2, π, and 3π/2) and label the endpoints of each (quarter points).
For each of these five numbers, the corresponding cosine values are 1, 0, -1, 0, 1.
Plot the (x, cos x) pairs and connect them with a smooth curve.
Graph may be extended left and right.

28. Figure: y = cos x, 0 < x < 2π

29. Figure: y = cos x, -∞ ≤ x ≤ ∞

30. Summary Basic Cosine Curve

31. Example:
True or False: The graph of y = cos x is symmetric with respect to the x – axis. Explain. False, the graph of y = cos x is symmetric with respect to the y-axis and an even function.

32. Your Turn:
True or False: The domain of the graph of y = cos x is the set of all real numbers. Explain.

33. Amplitude
In the rest of this section, you will study the graphic effect of each of the constants a, b, c, and d in equations of the forms y = d + a sin(bx – c) and y = d + a cos(bx – c).

34. Amplitude
The amplitude of a periodic function is half the difference between the maximum and minimum values.
The graph of y = a sin x or y = a cos x, with a  0, will have the same shape as the graph of y = sin x or y = cos x, respectively, except the range [|a|, |a|]. The amplitude is |a|.

35. Amplitude - Examples
Amplitude = (vertical distance between peak and trough)/2
= 2×amplitude

36. Amplitude
The constant factor a in y = a sin x acts as a scaling factor—a vertical stretch or vertical shrink of the basic sine curve. When | a | > 1, the basic sine curve is stretched, and when | a | < 1, the basic sine curve is shrunk.
The result is that the graph of y = a sin x ranges between –a and a instead of between –1 and 1.
The absolute value of a is the amplitude of the function y = a sin x. The range of the function y = a sin x for a  0 is –a  y  a.

37. Amplitude

38. Example: Compare graph of y = 3 sin x with graph of y = sin x.
Both graphs will have a period of 2π.
All the “y” values on y = a sin x, will be multiplied by “a”, compared with the “y” values on y = sin x.
The amplitude of y = a sin x will be:
| a | instead of “1” (the graph will be vertically stretched or squeezed depending on whether | a | > 1 or | a | < 1).
If a < 0, the graph of y = a sin x will be inverted with respect to y = sin x.
Example: Compare the graphs of:
y = 3 sin x and y = sin x

39. Example: Compare graph of y = 3 sin x with graph of y = sin x.
Make a table of values. 3 3
3sin x 1 1
sin x 2
3/2 
/2 x

40. Example: Compare graph of y = 3 sin x with graph of y = sin x.

41. Your Turn: Graph y = ½ cos x
Sketch the graph of y = ½ cos x.

42. Reflection
The graph of the function y = -a sin x and y = -a cos x is reflected about the x-axis compared to the graph y = a sin x and y = a cos x .
The negative of the function, y = -f(x), is a reflection about the x-axis.

43. Sketch the graph of y = -sin (x).
Example: Reflection
Sketch the graph of y = -sin (x).
The graph of y = -sin (x) is the graph of y = sin x reflected in the x-axis.
y = -sin (x) y x 2π π
y = sin x

44. Sketch the graph of y = -cos (x).
Example: Reflection
Sketch the graph of y = -cos (x).
The graph of y = -cos (x) is the graph of y = cos x reflected in the x-axis.
y = -cos (x) y x 2π π
y = cos x

45. Example:
Graph y = −2 cos x over the interval [−2π, 2π]. Find its amplitude and range.
Solution:
Make a table of values (the 5 key points).
Begin with the graph of y = cos x and multiply the
y-coordinate of each point on this graph by 2 to get
the graph of y = 2 cos x.
Next, reflect the graph in the x-axis to produce the graph of y =  −2 cos x.
The range of y = –2 cos x is [–2, 2] and its amplitude is |–2| = 2.

46. Example: Solution continued
Make a table of values. x
π/2 π
3π/2 2π
cos x 1 -1
2cos x 2 -2
-2cos x

47. Your Turn: Graph y = -1/3 sin x
Graph y = -1/3 sin x over the interval [0, 2π].