4.5 Graphs of Tangent, Cotangent, Secant, and Cosecant€¦ · 396 CHAPTER 4 Trigonometric Functions 4.5 Graphs of Tangent, Cotangent, Secant, and Cosecant The Tangent Function The

396 CHAPTER 4 Trigonometric Functions

4.5Graphs of Tangent, Cotangent, Secant, and Cosecant

The Tangent FunctionThe graph of the tangent function is shown below. As with the sine and cosinegraphs, this graph tells us quite a bit about the function’s properties. Here is asummary of tangent facts:

What you’ll learn about■ The Tangent Function■ The Cotangent Function■ The Secant Function■ The Cosecant Function

. . . and whyThis will give us functions for theremaining trigonometric ratios.

We now analyze why the graph of f !x" ! tan x behaves the way it does. It follows fromthe definitions of the trigonometric functions (Section 4.2) that

tan x ! "csoins

xx

".

Unlike the sinusoids, the tangent function has a denominator that might be zero,which makes the function undefined. Not only does this actually happen, but it hap-pens an infinite number of times: at all the values of x for which cos x ! 0. That iswhy the tangent function has vertical asymptotes at those values (Figure 4.45). Thezeros of the tangent function are the same as the zeros of the sine function: all the inte-ger multiples of ! (Figure 4.46).

Because sin x and cos x are both periodic with period 2!, you might expect the periodof the tangent function to be 2! also. The graph shows, however, that it is !.

FIGURE 4.46 The tangent functionhas zeros at the zeros of sine.

321

–3

y

x–2 π π

THE TANGENT FUNCTION

f !x" ! tan xDomain: All reals except odd multiples of !#2Range: All realsContinuous (i.e., continuous on its domain)Increasing on each interval in its domainSymmetric with respect to the origin (odd).Not bounded above or belowNo local extremaNo horizontal asymptotesVertical asymptotes: x ! k • !!#2" for all odd integers kEnd behavior: lim

x→$%tan x and lim

x→%tan x do not exist. (The function values

continually oscillate between $% and % and approach no limit.)

by [–4, 4]3 π /2]/2, [–3π

FIGURE 4.45 The tangent functionhas asymptotes at the zeros of cosine.

32

–3

y

x–2 π π

5144_Demana_Ch04pp349-442 01/12/06 11:06 AM Page 396

iii i

SECTION 4.5 Graphs of Tangent, Cotangent, Secant, and Cosecant 397

ALERTRemind students to pay close attentionto the radian and degree mode settings ontheir calculators. They should alwayscheck to see if the calculator mode isproperly set.

FIGURE 4.48 The cotangent hasasymptotes at the zeros of the sine function.

321

–3

y

x–2 π

FIGURE 4.47 The graph of (a) y ! tan 2xis reflected across the x-axis to produce thegraph of (b) y ! $tan 2x. (Example 1)

(b)

by [–4, 4]π ],[–π

(a)

by [–4, 4]π ],[–π

FIGURE 4.49 The cotangent haszeros at the zeros of the cosine function.

32

–3

y

x2 π–2 π

The constants a, b, h, and k influence the behavior of y ! a tan !b!x $ h"" & k in muchthe same way that they do for the graph of y ! a sin !b!x $ h"" & k. The constant ayields a vertical stretch or shrink, b affects the period, h causes a horizontal translation,and k causes a vertical translation. The terms amplitude and phase shift, however, arenot used, as they apply only to sinusoids.

EXAMPLE 1 Graphing a Tangent FunctionDescribe the graph of the function y ! $tan 2x in terms of a basic trigonometricfunction. Locate the vertical asymptotes and graph four periods of the function.

SOLUTION The effect of the 2 is a horizontal shrink of the graph of y ! tan x by afactor of 1#2, while the effect of the $1 is a reflection across the x-axis. Since the ver-tical asymptotes of y ! tan x are all odd multiples of !#2, the shrink factor causes thevertical asymptotes of y ! tan 2x to be all odd multiples of !#4 (Figure 4.47a). Thereflection across the x-axis (Figure 4.47b) does not change the asymptotes.

Since the period of the function y ! tan x is !, the period of the function y ! $tan 2xis (thanks again to the shrink factor) !#2. Thus, any window of horizontal length 2! willshow four periods of the graph. Figure 4.47b uses the window #$!, !$ by #$4, 4$.

Now try Exercise 5.

The other three trigonometric functions (cotangent, secant, and cosecant) are reciprocalsof tangent, cosine, and sine, respectively. (This is the reason that you probably do nothave buttons for them on your calculators.) As functions they are certainly interesting, butas basic functions they are unnecessary—we can do our trigonometric modeling andequation-solving with the other three. Nonetheless, we give each of them a brief sectionof its own in this book.

The Cotangent FunctionThe cotangent function is the reciprocal of the tangent function. Thus,

cot x ! "csoins

xx

".

The graph of y ! cot x will have asymptotes at the zeros of the sine function(Figure 4.48) and zeros at the zeros of the cosine function (Figure 4.49).

FOLLOW-UPStudents will be able to generate thegraphs for the tangent, cotangent, secant,and cosecant functions and to explore var-ious transformations of these graphs.

MOTIVATEAsk students to use a grapher to graph y ! tan x in both dot mode and connectedmode and ask them what observationsthey can make about the graph.



COTANGENT ON THE CALCULATOR

If your calculator does not have a “cotan”button, you can use the fact that cotan-gent and tangent are reciprocals. Forexample, the function in Example 2 canbe entered in a calculator as y !3/tan (x/2) & 1 or as y ! 3(tan (x/2))$1

& 1. Remember that it can not beentered as y ! 3 tan$1(x/2) & 1. (The $1exponent in that position represents afunction inverse, not a reciprocal.)

FIGURE 4.51 Characteristics of thesecant function are inferred from the fact thatit is the reciprocal of the cosine function.

32

–1–2–3

y

x2 π–2 π π

FIGURE 4.50 Two periods of f (x) ! 3 cot (x#2) & 1. (Example 2)

by [–10, 10]2 π ],[–2π

FIGURE 4.52 The graphs of y ! sec x and y ! $2 cos x. (Exploration 1)

[–6.5, 6.5] by [–3, 3]

EXPLORATION 1 Proving a Graphical Hunch

Figure 4.52 shows that the graphs of y ! sec x and y ! $2 cos x never seem tointersect.

If we stretch the reflected cosine graph vertically by a large enough number, willit continue to miss the secant graph? Or is there a large enough (positive) valueof k so that the graph of y ! sec x does intersect the graph of y ! $k cos x?

1. Try a few other values of k in your calculator. Do the graphs intersect?The graphs do not seem to intersect.

2. Your exploration should lead you to conjecture that the graphs of y ! sec xand y ! $k cos x will never intersect for any positive value of k. Verify thisconjecture by proving algebraically that the equation

$k cos x ! sec x

has no real solutions when k is a positive number.

EXPLORATION EXTENSIONSRepeat steps 1 and 4 with y ! csc x andy ! $k sin x.

EXAMPLE 2 Graphing a Cotangent FunctionDescribe the graph of f !x" ! 3 cot !x#2" & 1 in terms of a basic trigonometric func-tion. Locate the vertical asymptotes and graph two periods.

SOLUTION The graph is obtained from the graph of y ! cot x by effecting a hori-zontal stretch by a factor of 2, a vertical stretch by a factor of 3, and a vertical trans-lation up 1 unit. The horizontal stretch makes the period of the function 2! (twice theperiod of y ! cot x", and the asymptotes are at the even multiples of !. Figure 4.50shows two periods of the graph of f .

Now try Exercise 9.

The Secant FunctionImportant characteristics of the secant function can be inferred from the fact that it isthe reciprocal of the cosine function.

Whenever cos x ! 1, its reciprocal sec x is also 1. The graph of the secant function hasasymptotes at the zeros of the cosine function. The period of the secant function is 2!,the same as its reciprocal, the cosine function.

The graph of y ! sec x is shown with the graph of y ! cos x in Figure 4.51. A localmaximum of y ! cos x corresponds to a local minimum of y ! sec x, while a localminimum of y ! cos x corresponds to a local maximum of y ! sec x.



ONGOING ASSESSMENTSelf-Assessment: Ex. 5, 9, 29, 39Embedded Assessment: Ex. 41, 42, 44,59, 60

FIGURE 4.54 Characteristics of the cosecant function are inferred from the fact that it isthe reciprocal of the sine function.

321

–3

y

x–2 π

EXAMPLE 3 Solving a Trigonometric Equation Algebraically

Find the value of x between ! and 3!#2 that solves sec x ! $2.

SOLUTION We construct a reference triangle in the third quadrant that has theappropriate ratio, hyp#adj, equal to $2. This is most easily accomplished by choosingan x-coordinate of $1 and a hypotenuse of 2 (Figure 4.53a). We recognize this as a30'–60'–90' triangle that determines an angle of 240', which converts to 4!#3 radians(Figure 4.53b).

Therefore the answer is 4!#3.

FIGURE 4.53 A reference triangle in the third quadrant (a) with hyp#adj ! $2 determinesan angle (b) of 240 degrees, which converts to 4!#3 radians (Example 3).

Now try Exercise 29.

The Cosecant FunctionImportant characteristics of the cosecant function can be inferred from the fact that itis the reciprocal of the sine function.

Whenever sin x ! 1, its reciprocal csc x is also 1. The graph of the cosecant func-tion has asymptotes at the zeros of the sine function. The period of the cosecantfunction is 2!, the same as its reciprocal, the sine function.

The graph of y ! csc x is shown with the graph of y ! sin x in Figure 4.54. A local max-imum of y ! sin x corresponds to a local minimum of y ! csc x, while a local mini-mum of y ! sin x corresponds to a local maximum of y ! csc x.

240°

(b)

y

x

(a)

–1

2

y

x

FOLLOW-UPAsk what the relationship is between thedomain of the tangent and secant functionsand the zeros of the cosine function. (Thedomain of the tangent and secant functionsis the set of all real numbers that are notzeros of the cosine function.)

NOTES ON EXERCISESEx. 1–28 give students continued practicewith function transformations and shouldbe doable without graphing calculatorsEx. 29–34 anticipate, but do not use,inverse trigonometric functions.Ex. 51–56 provide practice with standard-ized test questions


o o oetc 45 ref 2 30 ref 2 60E 2 2i i fo o ous 30 60i 53 I

Q3 H

TAn o0 60ie o 1800 600av 2400 T

Tso

4z


EXAMPLE 4 Solving a Trigonometric Equation GraphicallyFind the smallest positive number x such that x2 ! csc x.

SOLUTION There is no algebraic attack that looks hopeful, so we solve this equa-tion graphically. The intersection point of the graphs of y ! x2 and y ! csc x that hasthe smallest positive x-coordinate is shown in Figure 4.55. We use the grapher todetermine that x % 1.068.

FIGURE 4.55 A graphical solution of a trigonometric equation. (Example 3)

Now try Exercise 39.

To close this section, we summarize the properties of the six basic trigonometric functionsin tabular form. The “n” that appears in several places should be understood as taking onall possible integer values: 0, ( 1, ( 2, ( 3,….

[–6.5, 6.5] by [–3, 3]

ARE COSECANT CURVES PARABOLAS?

Figure 4.55 shows a parabola intersect-ing one of the infinite number of U-shaped curves that make up the graphof the cosecant function. In fact, theparabola intersects all of those curvesthat lie above the x-axis, since theparabola must spread out to cover theentire domain of y ! x2, which is allreal numbers! The cosecant curves donot keep spreading out, as they arehemmed in by asymptotes. That meansthat the U-shaped curves in the cose-cant function are not parabolas.

Summary: Basic Trigonometric Functions

Function Period Domain Range Asymptotes Zeros Even#Odd

sin x 2! All reals #$1, 1$ None n! Oddcos x 2! All reals #$1, 1$ None !#2 & n! Eventan x ! x ) !#2 & n! All reals x ! !#2 & n! n! Oddcot x ! x ) n! All reals x ! n! !#2 & n! Oddsec x 2! x ) !#2 & n! !$%, $1$ ! #1, %" x ! !#2 & n! None Evencsc x 2! x ) n! !$%, $1$ ! #1, %" x ! n! None Odd

QUICK REVIEW 4.5 (For help, go to Sections 1.2, 2.6, and 4.3.)

In Exercises 1–4, state the period of the function.1. y ! cos 2x ! 2. y ! sin 3x 2!#3

3. y ! sin "13

" x 6! 4. y ! cos "12

" x 4!

In Exercises 5–8, find the zeros and vertical asymptotes ofthe function.

5. y ! "xx

$&

34

" 6. y ! "xx

&$

51

"

7. y !"!x $x2&"!x

1& 2"" 8. y ! "

x!xx

&$

23""

In Exercises 9 and 10, tell whether the function is odd, even, or nei-ther.

9. y ! x2 & 4 even 10. y ! "1x

" odd

5. Zero: 3; asymptote: x ! $4 6. Zero: $5; asymptote: x ! 17. Zero: $1; asymptotes: x ! 2 and x ! $28. Zero: $2; asymptotes: x ! 0 and x ! 3

ASSIGNMENT GUIDEEx. 3–48, multiples of 3, 51–56 all

COOPERATIVE LEARNINGGroup Activity: Ex. 63


2sin

put iny put in y

Hit 2ndTrace

opt 5move spider tolowest point ofintersection

1.068

i


SECTION 4.5 EXERCISES

In Exercises 1–4, identify the graph of each function. Use your under-standing of transformations, not your graphing calculator.

1. Graphs of one period of csc x and 2 csc x are shown.

2. Graphs of two periods of 0.5 tan x and 5 tan x are shown.

3. Graphs of csc x and 3 csc 2x are shown.

10

42

–10

y

68

xπ–π

y1

y2

10

42

–4

–10

y

68

–6–8

xπ–π

y1

y2

10

42

y

68

xπ–π

y1y2

4. Graphs of cot x and cot !x $ 0.5" & 3 are shown.

In Exercises 5–12, describe the graph of the function in terms of abasic trigonometric function. Locate the vertical asymptotes and graphtwo periods of the function.

5. y ! tan 2x 6. y ! $cot 3x

7. y ! sec 3x 8. y ! csc 2x

9. y ! 2 cot 2x 10. y ! 3 tan !x#2"11. y ! csc !x#2" 12. y ! 3 sec 4x

In Exercises 13–16, match the trigonometric function with its graph.Then give the Xmin and Xmax values for the viewing window inwhich the graph is shown. Use your understanding of transformations,not your graphing calculator.

13. y ! $2 tan x 14. y ! cot x

15. y ! sec 2x 16. y ! $csc x

[?, ?] by [–10, 10](d)

[?, ?] by [–10, 10](c)

[?, ?] by [–10, 10](b)

[?, ?] by [–10, 10](a)

10

4

–10

y

68

–8

xπ–π

y1

y2


W HW 5 10 20,21 22,24


In Exercises 17–20, analyze each function for domain, range, continuity,increasing or decreasing behavior, symmetry, boundedness, extrema,asymptotes, and end behavior.

17. f !x" ! cot x 18. f !x" ! sec x

19. f !x" ! csc x 20. f !x" ! tan !x#2"

In Exercises 21–28, describe the transformations required to obtain thegraph of the given function from a basic trigonometric graph.

21. y ! 3 tan x 22. y ! $tan x

23. y ! 3 csc x 24. y ! 2 tan x

25. y ! $3 cot "12

" x 26. y ! $2 sec "12

" x

27. y ! $tan "!2

" x & 2 28. y ! 2 tan !x $ 2

In Exercises 29–34, solve for x in the given interval. You should be ableto find these numbers without a calculator, using reference triangles inthe proper quadrants.

29. sec x " 2, 0 * x * !#2 !#3

30. csc x ! 2, !#2 * x * ! 5!#6

31. cot x ! $&3', !#2 * x * ! 5!#6

32. sec x ! $&2', ! * x * 3!#2 5!#4

33. csc x ! 1, 2! * x * 5!#2 5!#2

34. cot x ! 1, $! * x * $!#2 $3!#4

In Exercises 35–40, use a calculator to solve for x in the given interval.

35. tan x ! 1.3, 0 * x * "!2

" x % 0.92

36. sec x ! 2.4, 0 * x * "!2

" x % 1.14

37.cot x ! $0.6, "32!" * x * 2! x % 5.25

38.csc x ! $1.5, ! * x * "32!" x % 3.87

39. csc x ! 2, 0 * x * 2! x % 0.52 or x % 2.62

40. tan x ! 0.3, 0 * x * 2! x % 0.29 or x % 3.43

41. Writing to Learn The figure shows a unit circle and an angle twhose terminal side is in quadrant III.

y

xπt –

t

x2 + y2 = 1P1(–a, –b)

P2(a, b)

(a) If the coordinates of point P2 are !a, b", explain why the coor-dinates of point P1 on the circle and the terminal side of anglet $ ! are !$a, $b".

(b) Explain why tan t ! "ba

".

(c) Find tan !t $ !", and show that tan t ! tan !t $ !".(d) Explain why the period of the tangent function is !.

(e) Explain why the period of the cotangent function is !.

42. Writing to Learn Explain why it is correct to say y ! tan x isthe slope of the terminal side of angle x in standard position. P ison the unit circle.

43. Periodic Functions Let f be a periodic function with period p.That is, p is the smallest positive number such that

f !x & p" ! f !x"

for any value of x in the domain of f. Show that the reciprocal 1#fis periodic with period p.

44. Identities Use the unit circle to give a convincing argument forthe identity.

(a) sin !t & !" ! $sin t

(b) cos !t & !" ! $cos t

(c) Use (a) and (b) to show that tan!t & !" ! tan t. Explain whythis is not enough to conclude that the period of tangent is !.

45. Lighthouse Coverage The Bolivar Lighthouse is located ona small island 350 ft from the shore of the mainland as shown inthe figure.

(a) Express the distance d as a function of the angle x.

(b) If x is 1.55 rad, what is d? % 16,831 ft

dx

350 ft

y

xx

xP(cos x, sin x)



46. Hot-Air Balloon A hot-air balloon over Albuquerque, NewMexico, is being blown due east from point P and traveling at aconstant height of 800 ft. The angle y is formed by the ground andthe line of vision from P to the balloon. This angle changes as theballoon travels.

(a) Express the horizontal distance x as a function of the angle y.x ! 800 cot y

(b) When the angle is !#20 rad, what is its horizontal distancefrom P? % 5051 ft

(c) An angle of !#20 rad is equivalent to how many degrees? 9'

In Exercises 47–50, find approximate solutions for the equation in theinterval $! + x + !.

47. tan x ! csc x % ( 0.905 48. sec x ! cot x % 0.666 or % 2.475

49. sec x ! 5 cos x 50. 4 cos x ! tan x

49. % ( 1.107 or % ( 2.034 50. % 1.082 or % 2.060

Standardized Test Questions51. True or False The function f !x" ! tan x is increasing on the

interval !$∞, ∞". Justify your answer.

52. True or False If x ! a is an asymptote of the secant function,then cot a ! 0. Justify your answer.

You should answer these questions without using a calculator.

53. Multiple Choice The graph of y ! cot x can be obtained by ahorizontal shift of the graph of y ! A

(A) $tan x. (B) $cot x. (C) sec x.

(D) tan x. (E) csc x.

54. Multiple Choice The graph of y ! sec x never intersects thegraph of y ! E

(A) x. (B) x2. (C) csc x.

(D) cos x. (E) sin x.

55. Multiple Choice If k ) 0, what is the range of the functiony ! k csc x? D

(A) #$k, k$ (B) !$k, k"(C) !$∞, $k" ∪ !k, ∞" (D) !$∞, $k$ ∪ #k, ∞"(E) !$∞, $1#k$ ∪ #1#k, ∞"

P

800 ft

Windblowingdue east

y

x

56. Multiple Choice The graph of y ! csc x has the same set ofasymptotes as the graph of y ! C

(A) sin x. (B) tan x. (C) cot x.

(D) sec x. (E) csc 2x.

ExplorationsIn Exercises 57 and 58, graph both f and g in the #$!, !$ by #$10, 10$viewing window. Estimate values in the interval #$!, !$ for whichf , g.

57. f !x" ! 5 sin x and g !x" ! cot x

58. f !x" ! $tan x and g !x" ! csc x

59. Writing to Learn Graph the function f !x" ! $cot x on theinterval !$!, !". Explain why it is correct to say that f is increas-ing on the interval !0, !", but it is not correct to say that f isincreasing on the interval !$!, !".

60. Writing to Learn Graph functions f !x" ! $sec x and

g !x" ! "x $ !

1!#2""

simultaneously in the viewing window #0, !$ by #$10, 10$.Discuss whether you think functions f and g are equivalent.

61. Write csc x as a horizontal translation of sec x.

62. Write cot x as the reflection about the x-axis of a horizontal trans-lation of tan x. cot x ! $tan (x $ !#2)

Extending the Ideas63. Group Activity Television Coverage A television camera

is on a platform 30 m from the point on High Street where theWorthington Memorial Day Parade will pass. Express the dis-tance d from the camera to a particular parade float as a functionof the angle x, and graph the function over the interval$!#2 + x + !#2.

Hig

h St

reet

Float

Camera

30 m

d

x



64. What’s In a Name? The word sine comes from the Latin wordsinus, which means “bay” or “cove.” It entered the languagethrough a mistake (variously attributed to Gerardo of Cremona orRobert of Chester) in translating the Arabic word “jiba” (chord)as if it were “jaib” (bay). This was due to the fact that the Arabsabbreviated their technical terms, much as we do today. Imaginesomeone unfamiliar with the technical term “cosecant” trying toreconstruct the English word that is abbreviated by “csc.” It mightwell enter their language as their word for “cascade.”

The names for the other trigonometric functions can all beexplained.

(a) Cosine means “sine of the complement.” Explain why this is alogical name for cosine.

(b) In the figure below, BC is perpendicular to OC, which is aradius of the unit circle. By a familiar geometry theorem, BCis tangent to the circle. OB is part of a secant that intersectsthe unit circle at A. It lies along the terminal side of an angleof t radians in standard position. Write the coordinates of A asfunctions of t. (cos t, sin t )

(c) Use similar triangles to find length BC as a trig function of t.tan t

(d) Use similar triangles to find length OB as a trig function of t.sec t

(e) Use the results from parts (a), (c), and (d) to explain wherethe names “tangent, cotangent, secant,” and “cosecant”came from.

65. Capillary Action A film of liquid in a thin (capillary) tube has surface tension #(gamma) given by

# ! "12

" h$gr sec %,

y

x

1

t

A

B

CO D

where h is the height of the liquid in the tube, $ (rho) is the densi-ty of the liquid, g ! 9.8 m#sec2 is the acceleration due to gravity,r is the radius of the tube, and % (phi) is the angle of contactbetween the tube and the liquid’s surface. Whole blood has a sur-face tension of 0.058 N#m (newton per meter) and a density of1050 kg#m3. Suppose that blood rises to a height of 1.5 m in acapillary blood vessel of radius 4.7 - 10$6 m. What is the contactangle between the capillary vessel and the blood surface?!1 N ! 1 !kg • m"#sec2" % 0.8952 radians % 51.29'

66. Advanced Curve Fitting A researcher has reason to believethat the data in the table below can best be described by an alge-braic model involving the secant function:

y ! a sec !bx".Unfortunately, her calculator will only do sine regression. Sherealizes that the following two facts will help her:

"1y

" ! "a sec

1

!bx"" ! "

1a

" cos !bx"

and

cos !bx" ! sin (bx & "!2

").(a) Use these two facts to show that

"1y

" ! "1a

" sin (bx & "!2

").(b) Store the x values in the table in L1 in your calculator and the

y values in L2. Store the reciprocals of the y values in L3.Then do a sine regression for L3 !1#y" as a function of L1 !x".Write the regression equation.

(c) Use the regression equation in (b) to determine the values of aand b.

(d) Write the secant model: y ! a sec !bx". Does the curve fit the(L1, L2) scatter plot?

x 5 6 7 8

y 7.4359 9.2541 12.716 21.255

x 1 2 3 4

y 5.0703 5.2912 5.6975 6.3622

64. (a) If y ! sin t, then cosine is the sine of the angle that is the complement of t.(e) The length of BC is equal to tan t, and BC is tangent to the unit circle

at the initial side of angle t. The cotangent is the tangent of the anglethat is the complement of t. The length of OB is equal to sec t, and OBis part of a secant of the unit circle that is also the terminal side of t.The cosecant is the secant of the angle that is the complement of t.

66. (a) "1y

" ! "a se

1c(bx)" ! "

1a

" . "sec

1(bx)" ! "

1a

" cos(bx) ! "1a

" sin(bx & !/2)

(b) "1y

" ! 0.2sin("16

"x & "!

2")

(c) a ! 1/0.2 ! 5 and b ! 1/6


Documents

4.5 Graphs of Tangent, Cotangent, Secant, and Cosecant€¦ · 396 CHAPTER 4 Trigonometric Functions 4.5 Graphs of Tangent, Cotangent, Secant, and Cosecant The Tangent Function The