1. Graphs of the Sine and Cosine Functions

2. Graphing Trigonometric Functions
Graph in xy-plane
Write functions as
y = f(x) = sin x
y = f(x) = cos x
y = f(x) = tan x
y = f(x) = csc x
y = f(x) = sec x
y = f(x) = cot x
Variable x is an angle, measured in radians
Can be any real number

3. Graphing the Sine Function
Periodicity: Only need to graph on interval [0, 2¼] (One cycle)
Plot points and graph

4. Properties of the Sine Function
Domain: All real numbers
Range: [{1, 1]
Odd function
Periodic, period 2¼
x-intercepts: …, {2¼, {¼, 0, ¼, 2¼, 3¼, …
y-intercept: 0
Maximum value: y = 1, occurring at
Minimum value: y = {1, occurring at

5. Transformations of the Graph of the Sine Functions
Example.
Problem: Use the graph of y = sin x to graph
Answer:

6. Graphing the Cosine Function
Periodicity: Again, only need to graph on interval [0, 2¼] (One cycle)
Plot points and graph

7. Properties of the Cosine Function
Domain: All real numbers
Range: [{1, 1]
Even function
Periodic, period 2¼
x-intercepts:
y-intercept: 1
Maximum value: y = 1, occurring at
x = …, {2¼, 0, 2¼, 4¼, 6¼, …
Minimum value: y = {1, occurring at
x = …, {¼, ¼, 3¼, 5¼, …

8. Transformations of the Graph of the Cosine Functions
Example.
Problem: Use the graph of y = cos x to graph
Answer:

9. Sinusoidal Graphs
Graphs of sine and cosine functions appear to be translations of each other
Graphs are called sinusoidal
Conjecture.

10. Amplitude and Period of Sinusoidal Functions
Graphs of functions y = A sin x and y = A cos x will always satisfy inequality {jAj · y · jAj
Number jAj is the amplitude

11. Amplitude and Period of Sinusoidal Functions
Graphs of functions y = A sin x and y = A cos x will always satisfy inequality {jAj · y · jAj
Number jAj is the amplitude

12. Amplitude and Period of Sinusoidal Functions
Period of y = sin(!x) and y = cos(!x) is

13. Amplitude and Period of Sinusoidal Functions
Cycle: One period of y = sin(!x) or y = cos(!x)

14. Amplitude and Period of Sinusoidal Functions
Cycle: One period of y = sin(!x) or y = cos(!x)

15. Amplitude and Period of Sinusoidal Functions
Theorem. If ! > 0, the amplitude and period of y = Asin(!x) and y = Acos(! x) are given by
Amplitude = j Aj
Period =

16. Amplitude and Period of Sinusoidal Functions
Example.
Problem: Determine the amplitude and period of y = {2cos(¼x)
Answer:

17. Graphing Sinusoidal Functions
One cycle contains four important subintervals
For y = sin x and y = cos x these are
Gives five key points on graph

18. Graphing Sinusoidal Functions
Example.
Problem: Graph y = {3cos(2x)
Answer:

19. Finding Equations for Sinusoidal Graphs
Example.
Problem: Find an equation for the graph.
Answer:

20. Key Points Graphing Trigonometric Functions Graphing the Sine Function
Properties of the Sine Function
Transformations of the Graph of the Sine Functions
Graphing the Cosine Function
Properties of the Cosine Function
Transformations of the Graph of the Cosine Functions

21. Key Points (cont.) Sinusoidal Graphs
Amplitude and Period of Sinusoidal Functions
Graphing Sinusoidal Functions
Finding Equations for Sinusoidal Graphs

22. Graphs of the Tangent, Cotangent, Cosecant and Secant Functions
Section 5.5

23. Graphing the Tangent Function
Periodicity: Only need to graph on interval [0, ¼]
Plot points and graph

24. Properties of the Tangent Function
Domain: All real numbers, except odd multiples of
Range: All real numbers
Odd function
Periodic, period ¼
x-intercepts: …, {2¼, {¼, 0, ¼, 2¼, 3¼, …
y-intercept: 0
Asymptotes occur at

25. Transformations of the Graph of the Tangent Functions
Example.
Problem: Use the graph of y = tan x to graph
Answer:

26. Graphing the Cotangent Function
Periodicity: Only need to graph on interval [0, ¼]

27. Graphing the Cosecant and Secant Functions
Use reciprocal identities
Graph of y = csc x

28. Graphing the Cosecant and Secant Functions
Use reciprocal identities
Graph of y = sec x

29. Key Points Graphing the Tangent Function
Properties of the Tangent Function
Transformations of the Graph of the Tangent Functions
Graphing the Cotangent Function
Graphing the Cosecant and Secant Functions

30. Phase Shifts; Sinusoidal Curve Fitting

31. Graphing Sinusoidal Functions
y = A sin(!x), ! > 0
Amplitude jAj
Period
y = A sin(!x { Á)
Phase shift
Phase shift indicates amount of shift
To right if Á > 0
To left if Á < 0

32. Graphing Sinusoidal Functions
Graphing y = A sin(!x { Á) or y = A cos(!x { Á):
Determine amplitude jAj
Determine period
Determine starting point of one cycle:
Determine ending point of one cycle:

33. Graphing Sinusoidal Functions
Graphing y = A sin(!x { Á) or y = A cos(!x { Á):
Divide interval into four subintervals, each with length
Use endpoints of subintervals to find the five key points
Fill in one cycle

34. Graphing Sinusoidal Functions
Graphing y = A sin(!x { Á) or y = A cos(!x { Á):
Extend the graph in each direction to make it complete

35. Graphing Sinusoidal Functions
Example. For the equation
(a) Problem: Find the amplitude
Answer:
(b) Problem: Find the period
(c) Problem: Find the phase shift

36. Finding a Sinusoidal Function from Data
Example. An experiment in a wind tunnel generates cyclic waves. The following data is collected for 52 seconds. Let v represent the wind speed in feet per second and let x represent the time in seconds.
Time (in seconds), x
Wind speed (in feet per second), v 21 12 42 26 67 41 40 52 20

37. Finding a Sinusoidal Function from Data
Example. (cont.)
Problem: Write a sine equation that represents the data
Answer:

38. Key Points Graphing Sinusoidal Functions
Finding a Sinusoidal Function from Data