1. Graphs of Other Trigonometric Functions
6.5

2. Objectives Sketch the graphs of tangent functions.
Sketch the graphs of cotangent functions.
Sketch the graphs of secant and cosecant functions.

3. Graph of the Tangent Function

4. Graph of the Tangent Function
You know that the tangent function is odd. That is, tan(–x) = – tan x. Consequently, the graph of y = tan x is symmetric with respect to the origin. You also know from the identity tan x = sin x / cos x that the tangent is undefined for values at which cos x = 0. Two such values are x = ± / 2  ±

5. Graph of the Tangent Function
As indicated in the table, tan x increases without bound as x approaches  /2 from the left and decreases without bound as x approaches – /2 from the right.

6. Graph of the Tangent Function
So, the graph of y = tan x has vertical asymptotes at x =  /2 and x = – /2, as shown below.

7. Graph of the Tangent Function
Moreover, because the period of the tangent function is , vertical asymptotes also occur at x =  /2 + n, where n is an integer. The domain of the tangent function is the set of all real numbers other than x =  /2 + n, and the range is the set of all real numbers. Sketching the graph of y = a tan(bx – c) is similar to sketching the graph of y = a sin(bx – c) in that you locate key points that identify the intercepts and asymptotes.

8. Graph of the Tangent Function
Two consecutive vertical asymptotes can be found by solving the equations bx – c = and bx – c = The midpoint between two consecutive vertical asymptotes is an x-intercept of the graph. The period of the function y = a tan(bx – c) is the distance between two consecutive vertical asymptotes.

9. Graph of the Tangent Function
The amplitude of a tangent function is not defined. After plotting the asymptotes and the x-intercept, plot a few additional points between the two asymptotes and sketch one cycle. Finally, sketch one or two additional cycles to the left and right.

10. Tangent Functions

11. Steps for Sketching Graphs of Tangent
To graph y = tan bx, with b > 0, follow these steps.
Step 1 Determine the period, /b. To locate two adjacent vertical asymptotes solve the following equations for x:

12. Steps for Sketching Graphs of Tangent cont.
Step 2 Sketch the two vertical asymptotes found in Step 1.
Step 3 Divide the interval formed by the vertical asymptotes into four equal parts.
Step 4 Evaluate the function for the first-quarter point, midpoint, and third-quarter point, using the x-values found in Step 3.
Step 5 Join the points with a smooth curve, approaching the vertical asymptotes. Indicate additional asymptotes and periods of the graph as necessary.

13. Graph y = tan 2x
Step 1 The period of the function is /2. The two asymptotes have equations x = /4 and x = /4.
Step 2 Sketch two vertical asymptotes x =  /4.
Step 3 Divide the interval into four equal parts. This gives the following key x-values.
First quarter: /8 Middle value: 0 Third quarter: /8

14. Graph y = tan 2x continued
Step 4 Table of values
Step 5 Join the points with
a smooth curve, approaching the
vertical asymptotes. x
/8
/8 2x
/4
/4
tan 2x 1 1

15. Example – Sketching the Graph of a Tangent Function
Sketch the graph of y = tan . Solution: Step 1: By solving the equations x = – x =  you can see that two consecutive vertical asymptotes occur at x = – and x = . Step 2: Sketch two vertical asymptotes x =  .

16. Example – Solution
cont’d
Step 3: Divide the interval into four equal parts. First quarter: /2 Middle value: 0 Third quarter: /2 Step 4 : This gives the following key x-values.

17. Example – Solution
cont’d
Step 5 Join the points with a smooth curve, approaching the vertical asymptotes.

18. Graph y = 2 + tan x
Every y value for this function will be 2 units more than the corresponding y in y = tan x, causing the graph to be translated 2 units up compared to y = tan x.

19. Graph of the Cotangent Function

20. Graph of the Cotangent Function
The graph of the cotangent function is similar to the graph of the tangent function. It also has a period of . However, from the identity you can see that the cotangent function has vertical asymptotes when sin x is zero, which occurs at x = n, where n is an integer.

21. Graph of the Cotangent Function
The graph of the cotangent function is shown below. Note that two consecutive vertical asymptotes of the graph of y = a cot(bx – c) can be found by solving the equations bx – c = 0 and bx – c = .

22. Cotangent Functions

23. Steps for Sketching Graphs of Cotangent
To graph y = cot bx, with b > 0, follow these steps.
Step 1 Determine the period, /b. To locate two adjacent vertical asymptotes solve the following equations for x:

24. Steps for Sketching Graphs of Cotangent cont.
Step 2 Sketch the two vertical asymptotes found in Step 1.
Step 3 Divide the interval formed by the vertical asymptotes into four equal parts.
Step 4 Evaluate the function for the first-quarter point, midpoint, and third-quarter point, using the x-values found in Step 3.
Step 5 Join the points with a smooth curve, approaching the vertical asymptotes. Indicate additional asymptotes and periods of the graph as necessary.

25. Example – Sketching the Graph of a Cotangent Function
Sketch the graph of Solution: Step 1: By solving the equations = 0 =  x = 0 x = 3 you can see that two consecutive vertical asymptotes occur at x = 0 and x = 3 . Step 2: Sketch two vertical asymptotes x = 0 & 3.

26. Example – Solution
cont’d
Step 3: Divide the interval into four equal parts. First quarter: 3/4 Middle value: 3π/2 Third quarter: 9/4 Step 4 : This gives the following key x-values.

27. Example – Solution
cont’d
Step 5 Join the points with a smooth curve, approaching the vertical asymptotes.

28. Graphs of the Reciprocal Functions

29. Graphs of the Reciprocal Functions
You can obtain the graphs of the two remaining trigonometric functions from the graphs of the sine and cosine functions using the reciprocal identities csc x = and sec x = For instance, at a given value of x, the y-coordinate of sec x is the reciprocal of the y-coordinate of cos x. Of course, when cos x = 0, the reciprocal does not exist. Near such values of x, the behavior of the secant function is similar to that of the tangent function.

30. Graphs of the Reciprocal Functions
In other words, the graphs of tan x = and sec x = have vertical asymptotes where cos x = 0—that is, at x =  /2 + n, where n is an integer. Similarly, cot x = and csc x = have vertical asymptotes where sin x = 0—that is, at x = n, where n is an integer.

31. Cosecant Function

32. Secant Function

33. Graphs of the Reciprocal Functions
To sketch the graph of a secant or cosecant function, you should first make a sketch of its reciprocal function. For instance, to sketch the graph of y = csc x, first sketch the graph of y = sin x. Then take reciprocals of the y-coordinates to obtain points on the graph of y = csc x.

34. Graphs of the Reciprocal Functions
You can use this procedure to obtain the graphs shown below.

35. Graphs of the Reciprocal Functions
In comparing the graphs of the cosecant and secant functions with those of the sine and cosine functions, respectively, note that the “hills” and “valleys” are interchanged.

36. Graphs of the Reciprocal Functions
For instance, a hill (or maximum point) on the sine curve corresponds to a valley (a relative minimum) on the cosecant curve, and a valley (or minimum point) on the sine curve corresponds to a hill (a relative maximum) on the cosecant curve, as shown. Additionally, x-intercepts of the sine and cosine functions become vertical asymptotes of the cosecant and secant functions, respectively.

37. Steps for Sketching Graphs of Csc and Sec
To graph y = csc bx or y = sec bx, with b > 0, follow these steps.
Step 1 Graph the corresponding reciprocal function as a guide, using a dashed curve.
y = cos bx
y = a sec bx
y = a sin bx
y = a csc bx
Use as a Guide
To Graph

38. Steps for Sketching Graphs of Csc and Sec
Step 2 Sketch the vertical asymptotes. They will have equations of the form x = k, where k is an x-intercept of the graph of the guide function.
Step 3 Sketch the graph of the desired function by drawing the typical U-shapes branches between the adjacent asymptotes. The branches will be above the graph of the guide function when the guide function values are positive and below the graph of the guide function when the guide function values are negative.

39. Graph y = 2 sec x/2
Step 1 Graph the corresponding reciprocal function y = 2 cos x/2.
The function has amplitude 2 and one period of the graph lies along the interval that satisfies the inequality
Divide the interval into four equal parts.

40. Graph y = 2 sec x/2 continued
Step 2: Sketch the vertical asymptotes. These occur at x-values for which the guide function equals 0, such as x = 3, x = 3, x = ,
x = 3.

41. Graph y = 2 sec x/2 continued
Step 3: Sketch the graph of y = 2 sec x/2 by drawing the typical U-shaped branches, approaching the asymptotes.

42. Example – Sketching the Graph of a Cosecant Function
Sketch the graph of y = 2 csc Solution: Step 1: Begin by sketching the graph of y = 2 sin For this function, the amplitude is 2 and the period is 2.

43. Example – Solution
cont’d
By solving the equations x = x = you can see that one cycle of the sine function corresponds to the interval from x = – /4 to x = 7 /4.

44. Example – Solution
cont’d
The graph of this sine function is represented by the gray curve.

45. Example – Solution
cont’d
Step 2: Because the sine function is zero at the midpoint and endpoints of this interval, the corresponding cosecant function has vertical asymptotes at x = – /4, x = 3 /4, x = 7 /4 and so on.

46. Example – Solution
Step 3: The graph of the cosecant function is represented by the black curve.

47. Summary Trigonometric Graphs
Below is a summary of the characteristics of the six basic trigonometric functions.

48. Summary Trigonometric Graphs
cont’d

49. Assignment
Section 6.5 Notes Worksheet Sec. 6.5, Pg. 501; #1 – 29 odd