1. Trigonometric Functions of Real Numbers 5

2. More Trigonometric Graphs 5.4

3. More Trigonometric Graphs In this section, we graph: The tangent, cotangent, secant, and cosecant functions Transformations of these functions

4. Graphs of the Tangent, Cotangent, Secant, and Cosecant Function

5. Graphs of Tan, Cotan, Sec, and Csc Function We begin by stating the periodic properties of these functions. Recall that sine and cosine have period 2π. Since cosecant and secant are the reciprocals of sine and cosine, respectively, they also have period 2π. Tangent and cotangent, however, have period π.

6. Periodic Properties The functions tangent and cotangent have period π : The functions cosecant and secant have period 2π : tan(x + π) = tan x cot(x + π) = cot x csc(x + 2π) = csc x sec(x + 2π) = sec x

7. Graph of Tangent Function We first sketch the graph of tangent. Since it has period π, we need only sketch the graph on any interval of length π and then repeat the pattern to the left and to the right.

8. Graph of Tangent Function We sketch the graph on the interval (–π/2, π/2). Since tan π/2 and tan (–π/2) aren’t defined, we need to be careful in sketching the graph at points near π/2 and –π/2. As x gets near π/2 through values less than π/2, the value of tan x becomes large.

9. Graph of Tangent Function To see this, notice that, as x gets close to π/2, cos x approaches 0 and sin x approaches 1. Thus, tan x = sin x/cos x is large.

10. Graph of Tangent Function Here’s a table of values of tan x for x close to π/2 (≈ 1.570796).

11. Graph of Tangent Function Thus, by choosing x close enough to π/2 through values less than π/2, we can make the value of tan x larger than any given positive number. We express this by writing: This is read “tan x approaches infinity as x approaches π/2 from the left.”

12. Graph of Tangent Function Similarly, by choosing x close to –π/2 through values greater than –π/2, we can make tan x smaller than any given negative number. We write this as: This is read “tan x approaches negative infinity as x approaches –π/2 from the right.”

13. Graph of Tangent Function Thus, the graph of y = tan x approaches the vertical lines x = π/2 and x = –π/2. So, these lines are vertical asymptotes.

14. Graph of Tangent Function With the information we have so far, we sketch the graph of y = tan x for –π/2 < x < π/2.

15. Graph of Tangent Function The complete graph of tangent is now obtained using the fact that tangent is periodic with period π.

16. Graph of Cotangent Function By a similar analysis, the function y = cot x is graphed on the interval (0, π).

17. Graph of Cotangent Function Since cot x is undefined for x = nπ with n an integer, its complete graph has vertical asymptotes at these values.

18. Graphs of Cosecant and Secant Functions To graph the cosecant and secant functions, we use the reciprocal identities

19. Graph of Cosecant Function So, to graph y = csc x, we take the reciprocals of the y-coordinates of the points of the graph of y = sin x.

20. Graph of Secant Function Similarly, to graph y = sec x, we take the reciprocals of the y-coordinates of the points of the graph of y = cos x.

21. Graph of Cosecant Function Let’s consider more closely the graph of the function y = csc x on the interval 0 < x < π. We need to examine the values of the function near 0 and π. This is because, at these values, sin x = 0, and csc x is thus undefined.

22. Graph of Cosecant Function We see that: csc x → ∞ as x → 0 + csc x → ∞ as x → π – Thus, the lines x = 0 and x = π are vertical asymptotes.

23. Graph of Cosecant Function In the interval, π < x < 2π the graph is sketched in the same way. The values of csc x in that interval are the same as those in the interval 0 < x < π except for sign.

24. Graph of Cosecant Function The complete graph is now obtained from the fact that the function cosecant is periodic with period 2π. Note that the graph has vertical asymptotes at the points where sin x = 0—that is, at x = nπ, for n an integer.

25. Graph of Secant Function The 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 = (π/2) + nπ, for n an integer. So, the graph has vertical asymptotes at those points.

26. Graph of Secant Function Here’s the complete graph.

27. Graphs of Tan, Cotan, Sec, and Csc Function It is apparent that: The graphs of y = tan x, y = cot x, and y = csc x are symmetric about the origin. The graph of y = sec x is symmetric about the y-axis. This is because tangent, cotangent, and cosecant are odd functions, whereas secant is an even function.

28. Graphs Involving Tangent and Cotangent Functions

29. We now consider graphs of transformations of the tangent and cotangent functions.

30. E.g. 1—Graphing Tangent Curves Graph each function. (a) y = 2 tan x (b) y = –tan x We first graph y = tan x and then transform it as required.

31. E.g. 1—Graphing Tangent Curves To graph y = 2 tan x, we multiply the y-coordinate of each point on the graph of y = tan x by 2. Example (a)

32. E.g. 1—Graphing Tangent Curves The graph of y = –tan x is obtained from that of y = tan x by reflecting in the x-axis. Example (b)

33. Graphs Involving Tangent and Cotangent Functions The tangent and cotangent functions have period π. Thus, the functions y = a tan kx and y = a cot kx (k > 0) complete one period as kx varies from 0 to π, that is, for 0 ≤ kx ≤ π. Solving this inequality, we get 0 ≤ x ≤ π/k. So, they each have period π/k.

34. Tangent and Cotangent Curves The functions y = a tan kx and y = a cot kx (k > 0) have period π/k. Thus, one complete period of the graphs of these functions occurs on any interval of length π/k.

35. Tangent and Cotangent Curves To sketch a complete period of these graphs, it’s convenient to select an interval between vertical asymptotes: To graph one period of y = a tan kx, an appropriate interval is: To graph one period of y = a cot kx, an appropriate interval is:

36. E.g. 2—Graphing Tangent Curves Graph each function.

37. E.g. 2—Graphing Tangent Curves The period is π/2 and an appropriate interval is (–π/4, π/4). The endpoints x = –π/4 and x = π/4 are vertical asymptotes. Thus, we graph one complete period of the function on (–π/4, π/4). Example (a)

38. E.g. 2—Graphing Tangent Curves The graph has the same shape as that of the tangent function, but is shrunk horizontally by a factor of ½. We then repeat that portion of the graph to the left and to the right. Example (a)

39. E.g. 2—Graphing Tangent Curves The graph is the same as that in part (a). However, it is shifted to the right π/4. Example (b)

40. E.g. 3—A Shifted Cotangent Curve Graph. We first put this in the form y = a cot k(x – b) by factoring 3 from the expression : Thus, the graph is the same as that of y = 2 cot 3x, but is shifted to the right π/6.

41. E.g. 3—A Shifted Cotangent Curve The period of y = 2 cot 3x is π/3, and an appropriate interval is (0, π/3). To get the corresponding interval for the desired graph, we shift this interval to the right π/6. This gives:

42. E.g. 3—A Shifted Cotangent Curve Finally, we graph one period in the shape of cotangent on the interval (π/6, π/2) and repeat that portion of the graph to the left and to the right.

43. Graphs Involving the Cosecent and Secant Functions

44. Graphs Involving Cosecant and Secant Functions We have already observed that the cosecant and secant functions are the reciprocals of the sine and cosine functions. Thus, the following result is the counterpart of the result for sine and cosine curves in Section 5-3.

45. Cosecant and Secant Curves The functions y = a csc kx and y = a sec kx (k > 0) have period 2π/k. An appropriate interval on which to graph one complete period is [0, 2π/k].

46. E.g. 4—Graphing Cosecant Curves Graph each function.

47. E.g. 4—Graphing Cosecant Curves The period is 2π/2 = π. An appropriate interval is [0, π], and the asymptotes occur in this interval whenever sin 2x = 0. So, the asymptotes in this interval are: x = 0, x = π/2, x = π With this information, we sketch on the interval [0, π] a graph with the same general shape as that of one period of the cosecant function. Example (a)

48. E.g. 4—Graphing Cosecant Curves The complete graph is obtained by repeating this portion of the graph to the left and to the right. Example (a)

49. E.g. 4—Graphing Cosecant Curves We first write: From this, we see that the graph is the same as that in part (a), but shifted to the left π/4. Example (b)

50. E.g. 5—Graphing a Secant Curve Graph y = 3 sec ½x. The period is 2π ÷ ½ = 4π. An appropriate interval is [0, 4π], and the asymptotes occur in this interval wherever cos ½x = 0. Thus, the asymptotes in this interval are: x = π, x = 3π With this information, we sketch on the interval [0, 4π] a graph with the same general shape as that of one period of the secant function.

51. E.g. 5—Graphing a Secant Curve The complete graph is obtained by repeating this portion of the graph to the left and to the right.