1. More Trigonometric Graphs
5.4 – Day 1
More Trigonometric Graphs

2. Objectives Graphs of Tangent, Cotangent, Secant, and Cosecant
Graphs of Transformation of Tangent and Cotangent
Graphs of Transformations of Cosecant and Secant

3. More Trigonometric Graphs
In this section, we graph the tangent, cotangent, secant, and cosecant functions and transformations of these functions.

4. Graphs of Tangent and Cotangent

5. Graphs of Tangent and Cotangent
We begin by stating the periodic properties of these functions. Sine and cosine have period 2.
Since cosecant and secant are the reciprocals of sine and cosine, respectively, they also have period 2. Tangent and cotangent, however, have period .

6. Graphs of Tangent and Cotangent
Let’s use the values from the unit circle and the function y = tan x = sin x ÷ cos x to make a table of values and a graph of the tangent function.

7. Graphs of Tangent and Cotangent
The graph of y = tan x approaches the vertical lines x =  /2 and x = – /2. So, these lines are vertical asymptotes.
* Remember, the vertical asymptotes occur at the values that make the denominator (cos x) equal to 0.

8. Graphs of Tangent and Cotangent
With the information we have so far, we can sketch the “standard” graph of y = tan x from – /2 < x <  /2.
One period of y = tan x

9. Graphs of Tangent and Cotangent
The complete graph of tangent is now obtained using the fact that tangent is periodic with period .
y = tan x

10. Graphs of Tangent and Cotangent
Let’s use the values from the unit circle and the function y = cot x = cos x ÷ sin x to make a table of values and a graph of the cotangent function.

11. Graphs of Tangent and Cotangent
The function y = cot x is graphed on the interval (0, ).

12. Graphs of Tangent and Cotangent
Since cot x is undefined for x = n with n an integer, its complete graph has vertical asymptotes at these values.
One period y = cot x

13. Graphs of Transformations of
Tangent and Cotangent

14. Graphs of Transformations of Tangent and Cotangent
We now consider graphs of transformations of the tangent and cotangent functions.
Since the tangent and cotangent functions have period , the functions
y = a tan k(x-b) and y = a cot k(x-b) (k > 0)
complete one period as kx varies from 0 to , that is, for 0  kx  . Solving this inequality, we get 0  x   /k. So, they each have period  /k.
Remember, the value of “a” stretches or compresses the graph vertically.

15. Graphs of Transformations of Tangent and Cotangent
Thus, one complete period of the graphs of these functions occurs on any interval of length  /k.

16. Graphs of Transformations of Tangent and Cotangent
To sketch a complete period of these graphs, it’s convenient to select an interval between vertical asymptotes:
To find consecutive vertical asymptotes for the graph of
y = a tan k(x – b), solve the equations
k(x – b) = -π/ and k(x – b) = π/2 To find consecutive vertical asymptotes for the graph of
y = a cot k(x – b), solve the equations
k(x – b) = and k(x – b) = π

17. Example 2 – Graphing Tangent Curves
Graph the function:
(a) y = tan 2x

18. Example 2 – Graphing Tangent Curves
Graph the function:
(b) y = tan 2

19. Example 2 – Graphing Tangent Curves
(a) y = tan 2x (b) y = tan 2

20. Example 3 – A Shifted Cotangent Curve
Graph y = 2 cot

21. Example 3 – A Shifted Cotangent Curve

22. Graphs of Cosecant and Secant
It is apparent that the graphs of y = tan x and y = cot x are symmetric about the origin.
This is because tangent and cotangent are odd functions.

23. More Trigonometric Graphs
Practice: p
#1, 3, 5, 6, 11, 23, 35, 43, 53, 57

24. More Trigonometric Graphs
5.4 – Day 2
More Trigonometric Graphs

25. Objectives Graphs of Tangent, Cotangent, Secant, and Cosecant
Graphs of Transformation of Tangent and Cotangent
Graphs of Transformations of Cosecant and Secant

26. Graphs of Cosecant and Secant

27. Graphs of Cosecant and Secant
Recall that since cosecant and secant are the reciprocals of sine and cosine, respectively, they also have period 2.

28. Graphs of Cosecant and Secant
To graph the cosecant and secant functions, we use the reciprocal identities:
and
To graph y = csc x, we take the reciprocals of the y-coordinates of the points of the graph of y = sin x So, let’s do that!

29. Graphs of Cosecant and Secant
Graph y = csc x.

30. Graphs of Cosecant and Secant
One period of y = csc x

31. Graphs of Cosecant and Secant
The complete graph is obtained from the fact that the function cosecant is periodic with period 2. Note that the graph has vertical asymptotes at the points where sin x = 0, that is, at x = n, for n an integer.
y = csc x

32. Graphs of Cosecant and Secant
Similarly, to graph y = sec x, we take the reciprocals of the y-coordinates of the points of the graph of y = cos x.
So, let’s do that!

33. Graphs of Cosecant and Secant
Graph y = sec x.

34. Graphs of Cosecant and Secant
One period of y = sec x

35. Graphs of Cosecant and Secant
The complete graph of y = sec x is sketched in a similar manner. Observe that the domain of sec x is the set of all real numbers other than x = ( /2) + n, for n an integer, so the graph has vertical asymptotes at those points.
y = sec x

36. Graphs of Transformations of
Cosecant and Secant

37. Graphs of Transformations of Cosecant and Secant
An appropriate interval on which to graph one complete period is [0, 2 /k].

38. Graphs of Transformations of Cosecant and Secant
To find consecutive vertical asymptotes for the graph of y = a csc k(x – b), solve the equations k(x – b)= 0 and k(x – b)= π y = a sec k(x – b), solve the equations k(x – b)= -π/2 and k(x – b)= π/2

39. Example 4 – Graphing Cosecant Curves
Graph the function:

40. Example 4 – Graphing Cosecant Curves
Graph the function:

41. Example 4 – Graphing Cosecant Curves

42. Example 5 – Graphing a Secant Curve
Graph y = 3 sec

43. Example 5 – Graphing a Secant Curve
y = 3 sec

44. Graphs of Cosecant and Secant
It is apparent that the graph of y = csc x is symmetric about the origin, whereas that of y = sec x is symmetric about the y-axis.
This is because cosecant is an odd function, whereas secant is an even function.

45. More Trigonometric Graphs
Practice:
p. 405
#2, 4, 7, 8, 15, 17, 21, 25, 31, 33, 45, 51