what you should learn graph of the tangent function€¦ · sketch the graph of solution by solving...

332 Chapter 4 Trigonometry

What you should learn• Sketch the graphs of tangent

functions.

• Sketch the graphs of cotangent functions.

• Sketch the graphs of secantand cosecant functions.

• Sketch the graphs of dampedtrigonometric functions.

Why you should learn itTrigonometric functions canbe used to model real-life situations such as the distancefrom a television camera to a unitin a parade as in Exercise 76 onpage 341.

Graphs of Other Trigonometric Functions

Photodisc/Getty Images

4.6

Graph of the Tangent FunctionRecall that the tangent function is odd. That is, Consequently,the graph of is symmetric with respect to the origin. You also knowfrom the identity that the tangent is undefined for values atwhich Two such values are

As indicated in the table, tan increases without bound as approaches fromthe left, and decreases without bound as approaches from the right. So,the graph of has vertical asymptotes at and asshown in Figure 4.59. Moreover, because the period of the tangent function is vertical asymptotes also occur when where is an integer. Thedomain of the tangent function is the set of all real numbers other than

and the range is the set of all real numbers.

FIGURE 4.59

Sketching the graph of is similar to sketching the graph ofin that you locate key points that identify the intercepts and

asymptotes. Two consecutive vertical asymptotes can be found by solving theequations

and

The midpoint between two consecutive vertical asymptotes is an -intercept ofthe graph. The period of the function is the distance betweentwo consecutive vertical asymptotes. The amplitude of a tangent function is notdefined. After plotting the asymptotes and the -intercept, plot a few additionalpoints between the two asymptotes and sketch one cycle. Finally, sketch one ortwo additional cycles to the left and right.

x

y � a tan�bx � c�x

bx � c ��

2.bx � c � �

�

2

y � a sin�bx � c�y � a tan�bx � c�

1

2

3

−3

y = tan x

x

y

π2

3π π2

π2−π

23−

x � ��2 � n�,

nx � ��2 � n�,�,

x � ��2,x � ��2y � tan x��2x

��2xx

x � ±��2 � ±1.5708.cos x � 0.tan x � sin x�cos x

y � tan xtan��x� � �tan x.

PERIOD:DOMAIN: ALL

RANGE:VERTICAL ASYMPTOTES: x �

�2 � n�

��, ��x �

�2 � n�

�

x 0 1.5 1.57

Undef. 0 1 14.1 1255.8 Undef.�1�14.1�1255.8tan x

�

2�

4�

�

4�1.5�1.57�

�

2

333202_0406.qxd 12/8/05 8:43 AM Page 332

Sketching the Graph of a Tangent Function

Sketch the graph of

SolutionBy solving the equations

and

you can see that two consecutive vertical asymptotes occur at and Between these two asymptotes, plot a few points, including the -intercept,as shown in the table. Three cycles of the graph are shown in Figure 4.60.

Now try Exercise 7.

Sketching the Graph of a Tangent Function

Sketch the graph of


and

you can see that two consecutive vertical asymptotes occur at andBetween these two asymptotes, plot a few points, including the -inter-

cept, as shown in the table. Three cycles of the graph are shown in Figure 4.61.

Now try Exercise 9.

By comparing the graphs in Examples 1 and 2, you can see that the graph ofincreases between consecutive vertical asymptotes when

and decreases between consecutive vertical asymptotes when Inother words, the graph for is a reflection in the -axis of the graph for a > 0.xa < 0

a < 0.a > 0,y � a tan�bx � c�

xx � ��4.x � ��4

x ��

4x � �

�

4

2x ��

22x � �

�

2

y � �3 tan 2x.

xx � �.x � ��

x � �x � ��

x

2�

�

2

x

2� �

�

2

y � tan x

2.

Section 4.6 Graphs of Other Trigonometric Functions 333

Consider reviewing period, range, anddomain for all six trigonometricfunctions, especially emphasizing thedifference between the periods of thetangent and cotangent functions.

1

2

3

−3

y = tan x2y

xπ3π−π

FIGURE 4.60

6

−6

−4

−2

y = −3 tan 2x

π2

π2

−

y

xπ3π

4 4π34

π4

−−

FIGURE 4.61

Example 1

x 0

Undef. 3 0 Undef.�3�3 tan 2x

�

4�

8�

�

8�

�

4

x 0

Undef. 0 1 Undef.�1tan x2

��

2�

�

2��

Example 2

333202_0406.qxd 12/8/05 8:43 AM Page 333


Some graphing utilities have difficulty graphing trigonometricfunctions that have verticalasymptotes. Your graphing utilitymay connect parts of the graphsof tangent, cotangent, secant, andcosecant functions that are notsupposed to be connected. Toeliminate this problem, changethe mode of the graphing utilityto dot mode.

Techno logy

1

2

3

y = 2 cot

4−2

x3y

xππππ 63π

FIGURE 4.63

Graph of the Cotangent FunctionThe 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 is zero,which occurs at where is an integer. The graph of the cotangent func-tion is shown in Figure 4.62. Note that two consecutive vertical asymptotes of thegraph of can be found by solving the equations and

Sketching the Graph of a Cotangent Function

Sketch the graph of


and

you can see that two consecutive vertical asymptotes occur at and Between these two asymptotes, plot a few points, including the -intercept, asshown in the table. Three cycles of the graph are shown in Figure 4.63. Note thatthe period is the distance between consecutive asymptotes.

Now try Exercise 19.

3�,

xx � 3�.x � 0

x � 3�x � 0

x

3� 3�

x

3� 0

y � 2 cot x

3.

bx � c � �.bx � c � 0y � a cot�bx � c�

nx � n�,sin x

y � cot x �cos x

sin x

�.

ππ

1

2

3

y = cot x

22π32

−

y

xπ2π−π

FIGURE 4.62

PERIOD:DOMAIN: ALL

RANGE:VERTICAL ASYMPTOTES: x � n�

��, ��x � n�

�

x 0

Undef. 2 0 Undef.�22 cot x3

3�9�

43�

23�

4

Example 3

333202_0406.qxd 12/8/05 8:43 AM 34


Graphs of the Reciprocal FunctionsThe graphs of the two remaining trigonometric functions can be obtained fromthe graphs of the sine and cosine functions using the reciprocal identities

and

For instance, at a given value of the -coordinate of sec is the reciprocal ofthe -coordinate of cos Of course, when the reciprocal does notexist. Near such values of the behavior of the secant function is similar to thatof the tangent function. In other words, the graphs of

and

have vertical asymptotes at where is an integer, and the cosineis zero at these -values. Similarly,

and

have vertical asymptotes where —that is, at 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 first sketch the graph of Then take reciprocals of the -coordinates toobtain points on the graph of This procedure is used to obtain thegraphs shown in Figure 4.64.

In comparing the graphs of the cosecant and secant functions with those ofthe sine and cosine functions, note that the “hills” and “valleys” are interchanged.For example, 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) onthe sine curve corresponds to a hill (a relative maximum) on the cosecant curve,as shown in Figure 4.65. Additionally, -intercepts of the sine and cosinefunctions become vertical asymptotes of the cosecant and secant functions,respectively (see Figure 4.65).

x

y � csc x.yy � sin x.

y � csc x,

x � n�.sin x � 0

csc x �1

sin xcot x �

cos x

sin x

xnx � ��2 � n�,

sec x �1

cos xtan x �

sin x

cos x

x,cos x � 0,x.y

xyx,

sec x �1

cos x.csc x �

1

sin x

Using a graphing utility to graphcosecant and secant functions can helpreinforce the need to use the reciprocalfunctions.

3

4

2

1

−1

−2

−3

−4

πSine:maximum

Sine:minimum

Cosecant:relativemaximum

Cosecant:relativeminimum

π2

y

x

FIGURE 4.65

2

3

−1π

y = csc x

y = sin x

y

xπ−π

PERIOD:DOMAIN: ALL

RANGE:VERTICAL ASYMPTOTES:SYMMETRY: ORIGIN

FIGURE 4.64

x � n�

�1, ��, �1�x � n�

2�

x

3

2

−1

−2

−3

π ππ

y = sec x

y = cos x

2−

y

2π

PERIOD:DOMAIN: ALL

RANGE:VERTICAL ASYMPTOTES:SYMMETRY: -AXISy

x ��2 � n�

�1, ��, �1�x �

�2 � n�

2�

333202_0406.qxd 12/8/05 8:43 AM Page 335

Sketching the Graph of a Cosecant Function

Sketch the graph of

SolutionBegin by sketching the graph of

For this function, the amplitude is 2 and the period is By solving the equations

and

you can see that one cycle of the sine function corresponds to the interval fromto The graph of this sine function is represented by the

gray curve in Figure 4.66. Because the sine function is zero at the midpoint andendpoints of this interval, the corresponding cosecant function

has vertical asymptotes at etc. The graph ofthe cosecant function is represented by the black curve in Figure 4.66.


Sketching the Graph of a Secant Function

Sketch the graph of

SolutionBegin by sketching the graph of as indicated by the gray curve inFigure 4.67. Then, form the graph of as the black curve in the figure.Note that the -intercepts of

correspond to the vertical asymptotes

of the graph of Moreover, notice that the period of andis


�.y � sec 2xy � cos 2xy � sec 2x.

x �3�

4, . . .x �

�

4,x � �

�

4,

�3�

4, 0�, . . .��

4, 0�,��

�

4, 0�,

y � cos 2xxy � sec 2x

y � cos 2x,

y � sec 2x.

x � ��4, x � 3��4, x � 7��4,

� 2� 1sin�x � ��4��

y � 2 csc�x ��

4�

x � 7��4.x � ��4

x �7�

4x � �

�

4

x ��

4� 2�x �

�

4� 0

2�.

y � 2 sin�x ��

4�.

y � 2 csc�x ��

4�.


Example 4

Example 5

π

π

π

1

3

4

2

y = 2 csc x +4( ) πy = 2 sin x +

4( )y

x

FIGURE 4.66

π−π ππ22

y = sec 2x y = cos 2x

3

−2

−3

−1−

y

x

FIGURE 4.67

333202_0406.qxd 12/8/05 8:43 AM Page 336

Damped Trigonometric GraphsA product of two functions can be graphed using properties of the individualfunctions. For instance, consider the function

as the product of the functions and Using properties of absolutevalue and the fact that you have Consequently,

which means that the graph of lies between the lines andFurthermore, because

at

and

at

the graph of touches the line or the line at andhas -intercepts at A sketch of is shown in Figure 4.68. In the func-tion the factor is called the damping factor.

Damped Sine Wave

Sketch the graph of

SolutionConsider as the product of the two functions

and

each of which has the set of real numbers as its domain. For any real number you know that and So, which meansthat

Furthermore, because

at

and

at

the graph of touches the curves and at and has intercepts at A sketch is shown in Figure 4.69.


x � n��3.x � ��6 � n��3y � e�xy � �e�xf

x �n�

3f �x� � e�x sin 3x � 0

x ��

6�

n�

3f �x� � e�x sin 3x � ±e�x

�e�x ≤ e�x sin 3x ≤ e�x.

e�x sin 3x ≤ e�x,sin 3x ≤ 1.e�x ≥ 0x,

y � sin 3xy � e�x

f �x�

f �x� � e�x sin 3x.

xf �x� � x sin x,fx � n�.x

x � ��2 � n�y � xy � �xf

x � n�f �x� � x sin x � 0

x ��

2� n�f �x� � x sin x � ±x

y � x.y � �xf �x� � x sin x

�x ≤ x sin x ≤ x0 ≤ xsin x ≤ x.sin x ≤ 1,

y � sin x.y � x

f �x� � x sin x


Example 6

π

−ππ

π

π3

π−3

2

π−2

f(x) = x sin x

yy = −x = x

y

x

FIGURE 4.68

π ππ3 3

2

6

4

−4

−6

y = e−x

y = −e−x

f(x) = e−x sin 3x y

x

FIGURE 4.69

Do you see why the graph oftouches the lines

at andwhy the graph has -interceptsat Recall that the sinefunction is equal to 1 at

odd multiplesof and is equal to 0 at

multiples of ��.�3�, . . .2�,�,��2�

�5��2, . . . 3��2,��2,

x � n�?x

x � ��2 � n�y � ±xf �x� � x sin x

333202_0406.qxd 12/8/05 8:43 AM Page 337


Figure 4.70 summarizes the characteristics of the six basic trigonometric functions.

y = sin x

π π−π ππ−2 22

3

2

1

−2

y

x

DOMAIN: ALL REALS

RANGE:PERIOD: 2�

��1, 1�

y = cos x

π ππ− 2

2

−1

−2

y

x

DOMAIN: ALL REALS

RANGE:PERIOD: 2�

��1, 1�

y = tan x

π

2

3

1

y

xπ52

π2

3π2

π2−

DOMAIN: ALL

RANGE:PERIOD: �

��, ��x �

�2 � n�

y = csc x =

π2

2

3

1

1sin xy

x−π π π2

DOMAIN: ALL

RANGE:PERIOD:FIGURE 4.70

2�

�1, ��, �1�x � n�

π ππ

3

−2

−3

2−

y = sec x = 1cos xy

xπ2

3π2

− π2

DOMAIN: ALL

RANGE:PERIOD: 2�

�1, ��, �1�x �

�2 � n�

y x == cot

2

3

1

tan x1y

xπ π2

DOMAIN: ALL

RANGE:PERIOD: �

��, ��x � n�

W RITING ABOUT MATHEMATICS

Combining Trigonometric Functions Recall from Section 1.8 that functions can becombined arithmetically. This also applies to trigonometric functions. For each ofthe functions

and

(a) identify two simpler functions and that comprise the combination, (b) use atable to show how to obtain the numerical values of from the numerical valuesof and and (c) use graphs of and to show how may be formed.

Can you find functions

and

such that for all x?f �x� � g�x� � 0

g�x� � d � a cos�bx � c�f �x� � d � a sin�bx � c�

hgfg�x�,f �x�h�x�

gf

h�x� � cos x � sin 3xh�x� � x � sin x

333202_0406.qxd 12/8/05 8:43 AM Page 338


Exercises 4.6

In Exercises 1–6, match the function with its graph. Statethe period of the function. [The graphs are labeled (a), (b),(c), (d), (e), and (f).]

(a) (b)

(c) (d)

(e) (f)

1. 2.

3. 4.

5. 6.

In Exercises 7–30, sketch the graph of the function. Includetwo full periods.

7. 8.

9. 10.

11. 12.

13. 14.

15. 16.

17. 18.

19. 20.

21. 22.

23. 24.

25. 26.

27. 28.

29. 30.

In Exercises 31– 40, use a graphing utility to graph thefunction. Include two full periods.

31. 32.

33. 34.

35. 36.

37. 38.

39. 40. y �1

3 sec��x

2�

�

2�y � 0.1 tan��x

4�

�

4�y � 2 sec�2x � ��y � �csc�4x � ��

y �1

4 cot�x �

�

2�y � tan�x ��

4�y � sec �xy � �2 sec 4x

y � �tan 2xy � tan x

3

y � 2 cot�x ��

2�y �1

4 csc�x �

�

4�y � �sec� x � 1y � 2 sec�x � ��y � csc�2x � ��y � csc�� x�

y � tan�x � ��y � tan �x

4

y � �12 tan xy �

12 sec 2x

y � 3 cot �x

2y � cot

x

2

y � csc x

3y � csc

x

2

y � �2 sec 4x � 2y � sec �x � 1

y � 3 csc 4xy � csc �x

y �14 sec xy � �

12 sec x

y � �3 tan �xy � tan 3x

y �14 tan xy �

13 tan x

y � �2 sec �x

2y �

1

2 sec

�x

2

y � �csc xy �1

2 cot �x

y � tan x

2y � sec 2x

3

1

y

x

4

π2

y

x

−3

2

32 2

3

π π−

y

x

−3−4

12

2

34

π−2

3π

y

x

2

1

y

x

1

2

1

y

x

VOCABULARY CHECK: Fill in the blanks.

1. The graphs of the tangent, cotangent, secant, and cosecant functions all have ________ asymptotes.

2. To sketch the graph of a secant or cosecant function, first make a sketch of its corresponding ________ function.

3. For the functions given by is called the ________ factor of the function

4. The period of is ________.

5. The domain of is all real numbers such that ________.

6. The range of is ________.

7. The period of is ________.

PREREQUISITE SKILLS REVIEW: Practice and review algebra skills needed for this section at www.Eduspace.com.

y � csc x

y � sec x

y � cot x

y � tan x

f �x�.g�x�f �x� � g�x� � sin x,

333202_0406.qxd 12/8/05 8:43 AM Page 339

In Exercises 41– 48, use a graph to solve the equation on theinterval

41.

42.

43.

44.

45.

46.

47.

48.

In Exercises 49 and 50, use the graph of the function todetermine whether the function is even, odd, or neither.

49. 50.

51. Graphical Reasoning Consider the functions given by

and

on the interval

(a) Graph and in the same coordinate plane.

(b) Approximate the interval in which

(c) Describe the behavior of each of the functions as approaches How is the behavior of related to thebehavior of as approaches

52. Graphical Reasoning Consider the functions given by

and

on the interval

(a) Use a graphing utility to graph and in the sameviewing window.

(b) Approximate the interval in which

(c) Approximate the interval in which How doesthe result compare with that of part (b)? Explain.

In Exercises 53–56, use a graphing utility to graph the twoequations in the same viewing window. Use the graphs todetermine whether the expressions are equivalent. Verifythe results algebraically.

53.

54.

55.

56.

In Exercises 57– 60, match the function with its graph.Describe the behavior of the function as approacheszero. [The graphs are labeled (a), (b), (c), and (d).]

(a) (b)

(c) (d)

57.

58.

59.

60.

Conjecture In Exercises 61–64, graph the functions andUse the graphs to make a conjecture about the

relationship between the functions.

61.

62.

63.

64.

In Exercises 65–68, use a graphing utility to graph thefunction and the damping factor of the function in thesame viewing window. Describe the behavior of thefunction as increases without bound.

65. 66.

67. 68.

Exploration In Exercises 69–74, use a graphing utility tograph the function. Describe the behavior of the functionas approaches zero.

69. 70. x > 0y �4

x� sin 2x,x > 0y �

6

x� cos x,

x

h�x� � 2�x2�4 sin xf �x� � 2�x�4 cos �x

f �x� � e�x cos xg�x� � e�x2�2 sin x

x

g�x� �1

2�1 � cos � x�f �x� � cos2

�x

2,

g�x� �12�1 � cos 2x�f �x� � sin2 x,

g�x� � 2 sin xf �x� � sin x � cos�x ��

2�,

g�x� � 0f �x� � sin x � cos�x ��

2�,

g.f

g�x� � x cos x

g�x� � x sin x

f �x� � x sin x

f �x� � x cos x

ππ −1

2

−2

−

4

1

3

y

xππ

−4

2

−2−

4

y

x

π π

−4

2

2 23

4

y

x

−6

2

−1 π2−2

−3−4−5

y

x

x

y2 � tan2 xy1 � sec2 x � 1,

y2 � cot xy1 �cos x

sin x,

y2 � tan xy1 � sin x sec x,

y2 � 1y1 � sin x csc x,

2f < 2g.

f < g.

gf

��1, 1�.

g�x� �1

2 sec

�x

2f �x� � tan

�x

2

�?xfg�.

x

f > g.

gf

�0, ��.

g�x� �1

2 csc xf �x� � 2 sin x

f �x� � tan xf �x� � sec x

csc x � �23

3

csc x � 2

sec x � 2

sec x � �2

cot x � 1

cot x � �33

tan x � 3

tan x � 1

[�2�, 2� ].


333202_0406.qxd 12/8/05 8:43 AM Page 340


71. 72.

73. 74.

75. Distance A plane flying at an altitude of 7 miles above aradar antenna will pass directly over the radar antenna (seefigure). Let be the ground distance from the antenna tothe point directly under the plane and let be the angle ofelevation to the plane from the antenna. ( is positive as theplane approaches the antenna.) Write as a function of and graph the function over the interval

76. Television Coverage A television camera is on areviewing platform 27 meters from the street on which aparade will be passing from left to right (see figure). Writethe distance from the camera to a particular unit in theparade as a function of the angle and graph the functionover the interval (Consider asnegative when a unit in the parade approaches from theleft.)

78. Sales The projected monthly sales (in thousands ofunits) of lawn mowers (a seasonal product) are modeled by

where is the time (inmonths), with corresponding to January. Graph thesales function over 1 year.

79. Meterology The normal monthly high temperatures (in degrees Fahrenheit) for Erie, Pennsylvania are approximated by

and the normal monthly low temperatures are approxi-mated by

where is the time (in months), with correspondingto January (see figure). (Source: National Oceanic andAtmospheric Administration)

(a) What is the period of each function?

(b) During what part of the year is the difference betweenthe normal high and normal low temperatures greatest?When is it smallest?

(c) The sun is northernmost in the sky around June 21, butthe graph shows the warmest temperatures at a laterdate. Approximate the lag time of the temperaturesrelative to the position of the sun.

Month of year

Tem

pera

ture

(in

degr

ees

Fah

renh

eit)

1 2 3 4 5 6 7 8 9 10 11 12

20

40

60

80

t

H(t)

L(t)

t � 1t

L�t� � 39.36 � 15.70 cos � t

6� 14.16 sin

� t

6

L

H�t� � 54.33 � 20.38 cos � t

6� 15.69 sin

� t

6

H

t � 1tS � 74 � 3t � 40 cos��t�6�,

S

xd

Camera

27 m

Not drawn to scale

x��2 < x < ��2.x,

d

x

7 mi

d

Not drawn to scale

0 < x < �.xd

dx

d

h�x� � x sin 1

xf �x� � sin

1

x

f �x� �1 � cos x

xg�x� �

sin x

x

77. Predator-Prey Model The population of coyotes (apredator) at time (in months) in a region is estimatedto be

and the population of rabbits (its prey) is estimated tobe

R

C � 5000 � 2000 sin � t12

tC

Model It

Model It (cont inued)

(a) Use a graphing utility to graph both models in thesame viewing window. Use the window setting

(b) Use the graphs of the models in part (a) to explainthe oscillations in the size of each population.

(c) The cycles of each population follow a periodicpattern. Find the period of each model and describeseveral factors that could be contributing to thecyclical patterns.

0 ≤ t ≤ 100.

R � 25,000 � 15,000 cos � t

12.

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80. Harmonic Motion An object weighing pounds issuspended from the ceiling by a steel spring (see figure).The weight is pulled downward (positive direction) fromits equilibrium position and released. The resulting motionof the weight is described by the function

where is the distance (in feet) and is the time (in seconds).

(a) Use a graphing utility to graph the function.

(b) Describe the behavior of the displacement function forincreasing values of time

Synthesis

True or False? In Exercises 81 and 82, determine whetherthe statement is true or false. Justify your answer.

81. The graph of can be obtained on a calculator bygraphing the reciprocal of

82. The graph of can be obtained on a calculator bygraphing a translation of the reciprocal of

83. Writing Describe the behavior of as approaches from the left and from the right.

84. Writing Describe the behavior of as approaches from the left and from the right.

85. Exploration Consider the function given by

(a) Use a graphing utility to graph the function and verifythat there exists a zero between 0 and 1. Use the graphto approximate the zero.

(b) Starting with generate a sequence where For example,

What value does the sequence approach?

86. Approximation Using calculus, it can be shown that thetangent function can be approximated by the polynomial

where is in radians. Use a graphing utility to graph thetangent function and its polynomial approximation in thesame viewing window. How do the graphs compare?

87. Approximation Using calculus, it can be shown that thesecant function can be approximated by the polynomial

where is in radians. Use a graphing utility to graph thesecant function and its polynomial approximation in thesame viewing window. How do the graphs compare?

88. Pattern Recognition

(a) Use a graphing utility to graph each function.

(b) Identify the pattern started in part (a) and find afunction that continues the pattern one more term.Use a graphing utility to graph

(c) The graphs in parts (a) and (b) approximate the periodicfunction in the figure. Find a function that is a betterapproximation.

Skills Review

In Exercises 89–92, solve the exponential equation. Roundyour answer to three decimal places.

89. 90.

91. 92.

In Exercises 93–98, solve the logarithmic equation. Roundyour answer to three decimal places.

93. 94.

95. 96.

97.

98. log6 x � log6�x2 � 1� � log6 64x

log8 x � log8�x � 1� �13

ln x � 4 � 5ln�x2 � 1� � 3.2

ln�14 � 2x� � 68ln�3x � 2� � 73

�1 �0.15365 �

365t

� 5300

1 � e�x � 100

83x � 98e2x � 54

3

1

y

x

y4

y3.y3

y2 �4

��sin �x �1

3 sin 3�x �

1

5 sin 5�x�

y1 �4

��sin �x �1

3 sin 3�x�

x

sec x � 1 �x 2

2!�

5x4

4!

x

tan x � x �2x3

3!�

16x 5

5!

�

x3 � cos�x2�x2 � cos�x1�x1 � cos�x0�x0 � 1

xn � cos�xn�1�.x3, . . . ,x1, x2,x0 � 1,

f �x� � x � cos x.

�xf �x� � csc x

��2xf �x� � tan x

y � sin x.y � sec x

y � sin x.y � csc x

t.

Equilibrium

y

ty

t > 0y �1

2e�t�4 cos 4t,

W


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what you should learn graph of the tangent function€¦ · sketch the graph of solution by solving...

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