4.4 trigonometric functions of any angle. 312... · x 4 3 cos x r 3 5 sin y r 4 5 r x2 y2 3 2 42 25...

312 Chapter 4 Trigonometry

What you should learn• Evaluate trigonometric

functions of any angle.

• Use reference angles to evaluate trigonometricfunctions.

• Evaluate trigonometric functions of real numbers.

Why you should learn itYou can use trigonometric functions to model and solvereal-life problems. For instance,in Exercise 87 on page 319, youcan use trigonometric functionsto model the monthly normaltemperatures in New York Cityand Fairbanks, Alaska.

Trigonometric Functions of Any Angle

James Urbach/SuperStock

4.4

IntroductionIn Section 4.3, the definitions of trigonometric functions were restricted to acuteangles. In this section, the definitions are extended to cover any angle. If is anacute angle, these definitions coincide with those given in the preceding section.

Because cannot be zero, it follows that the sine and cosinefunctions are defined for any real value of However, if the tangent andsecant of are undefined. For example, the tangent of is undefined.Similarly, if the cotangent and cosecant of are undefined.

Evaluating Trigonometric Functions

Let be a point on the terminal side of Find the sine, cosine, andtangent of

Solution

Referring to Figure 4.36, you can see that and

So, you have the following.

Now try Exercise 1.

tan � �y

x� �

4

3

cos � �x

r� �

3

5

sin � �y

r�

4

5

r � �x2 � y2 � ��3�2 � 42 � �25 � 5.

y � 4,x � �3,

�.�.��3, 4�

�y � 0,90��

x � 0,�.r � �x2 � y2

�

Definitions of Trigonometric Functions of Any AngleLet be an angle in standard position with a point on the terminal sideof and

csc � �ry

, y � 0sec � �rx

, x � 0

cot � �xy

, y � 0tan � �yx

, x � 0

( , )x y

rθ

y

x

cos � �xr

sin � �yr

r � �x2 � y2 � 0.��x, y��

Example 1

( 3, 4)−

−1−2−3 1

1

2

3

4

r

θ

y

x

FIGURE 4.36

333202_0404.qxd 12/7/05 11:05 AM Page 312

Section 4.4 Trigonometric Functions of Any Angle 313

The signs of the trigonometric functions in the four quadrants can bedetermined easily from the definitions of the functions. For instance, because

it follows that is positive wherever which is inQuadrants I and IV. (Remember, is always positive.) In a similar manner, youcan verify the results shown in Figure 4.37.

Evaluating Trigonometric Functions

Given and find and

SolutionNote that lies in Quadrant IV because that is the only quadrant in which thetangent is negative and the cosine is positive. Moreover, using

and the fact that is negative in Quadrant IV, you can let and So,and you have

Now try Exercise 17.

Trigonometric Functions of Quadrant Angles

Evaluate the cosine and tangent functions at the four quadrant angles 0, and

SolutionTo begin, choose a point on the terminal side of each angle, as shown in Figure4.38. For each of the four points, and you have the following.


�x, y� � �0, �1� tan 3�

2�

y

x�

�1

0 ⇒ undefined cos

3�

2�

x

r�

0

1� 0

�x, y� � ��1, 0�tan � �y

x�

0

�1� 0cos � �

x

r�

�1

1� �1

�x, y� � �0, 1�tan �

2�

y

x�

1

0 ⇒ undefinedcos

�

2�

x

r�

0

1� 0

�x, y� � �1, 0�tan 0 �y

x�

0

1� 0cos 0 �

x

r�

1

1� 1

r � 1,

3�

2.

�

2, �,

� 1.6008.

sec � �r

x�

�41

4

� �0.7809

sin � �y

r�

�5�41

r � �16 � 25 � �41x � 4.y � �5y

� �5

4tan � �

y

x

�

sec �.sin �cos � > 0,tan � � �54

rx > 0,cos �cos � � x�r,

x > 0y > 0

x < 0y > 0

x < 0y < 0

x > 0y < 0

π π2

< <θ π2

0 < <θ

π π2

< < 2θ3ππ2

< <θ 3

y

x

x

Quadrant IQuadrant II

Quadrant III Quadrant IV

sin : +cos : +tan : +

θθθ

sin :cos : +tan :

−

−

θθθ

sin : +cos :tan :

−−

θθθ

sin :cos :tan : +

−−

θθθ

y

FIGURE 4.37

π2

π2

3

(0, 1)

π 0

(1, 0)(−1, 0)

(0, −1)

y

x

FIGURE 4.38

Example 2

Example 3

333202_0404.qxd 12/7/05 11:05 AM Page 313


Reference AnglesThe values of the trigonometric functions of angles greater than (or less than

) can be determined from their values at corresponding acute angles calledreference angles.

Figure 4.39 shows the reference angles for in Quadrants II, III, and IV.

FIGURE 4.39

Finding Reference Angles

Find the reference angle

a. b. c.

Solutiona. Because lies in Quadrant IV, the angle it makes with the -axis is

Degrees

Figure 4.40 shows the angle and its reference angle

b. Because 2.3 lies between and it follows that it isin Quadrant II and its reference angle is

Radians


c. First, determine that is coterminal with which lies in QuadrantIII. So, the reference angle is

Degrees



�� 45�.� �135��

� 45�.

�� 225� � 180�

225�,�135�

�� 2.3.� � 2.3

� 0.8416.

�� 2.3

� � 3.1416,��2 � 1.5708

�� 60�.� � 300�

� 60�.

�� 360� � 300�

x300�

� � �135�� 2.3� � 300�

��.

Quadrant IV

′ = 2 − (radians)′ = 360° − (degrees)

θReferenceangle: ′θ

θθ

πθθ

Quadrant III

′ = − (radians)′ = − 180° (degrees)

θReferenceangle: ′θ

θθ

πθθ

Quadrant II

Referenceangle: ′

′ = − (radians)′ = 180° − (degrees)

θ

θ

θθ

πθθ

�

0�90�

= 300°

′ = 60°θ

θ

y

x

FIGURE 4.40

= 2.3′ = − 2.3θ θπ

y

x

FIGURE 4.41

225° and −135°are coterminal.

= −135°

225°

′ = 45°θ θ

y

x

FIGURE 4.42

Sketching several angles with theirreference angles may help reinforce thefact that the reference angle is the acuteangle formed with the horizontal.

Definition of Reference AngleLet be an angle in standard position. Its reference angle is the acute angle

formed by the terminal side of and the horizontal axis.��

Example 4

333202_0404.qxd 12/7/05 11:05 AM 14

Trigonometric Functions of Real NumbersTo see how a reference angle is used to evaluate a trigonometric function,consider the point on the terminal side of as shown in Figure 4.43. Bydefinition, you know that

and

For the right triangle with acute angle and sides of lengths and youhave

and

So, it follows that and are equal, except possibly in sign. The same istrue for and and for the other four trigonometric functions. In allcases, the sign of the function value can be determined by the quadrant in which

lies.

By using reference angles and the special angles discussed in the precedingsection, you can greatly extend the scope of exact trigonometric values. Forinstance, knowing the function values of means that you know the functionvalues of all angles for which is a reference angle. For convenience, the tablebelow shows the exact values of the trigonometric functions of special angles andquadrant angles.

Trigonometric Values of Common Angles

30�30�

�

tan ��tan �sin ��sin �

tan �� opp

adj�

�y��x�.

sin �� opp

hyp�

�y�r

�y�,�x��

tan � �y

x.sin � �

y

r

�,�x, y�


(x, y)

′

opp

adj

r = hyp

θ θ

y

x

FIGURE 4.43opp � �y�, adj � �x�

Learning the table of values atthe right is worth the effortbecause doing so will increaseboth your efficiency and yourconfidence. Here is a pattern forthe sine function that may helpyou remember the values.

Reverse the order to get cosinevalues of the same angles.

�42

�32

�22

�12

�02

sin �

90�60�45�30�0��

Evaluating Trigonometric Functions of Any AngleTo find the value of a trigonometric function of any angle

1. Determine the function value for the associated reference angle

2. Depending on the quadrant in which lies, affix the appropriate sign tothe function value.

�

��.

�:

(degrees)

(radians) 0

0 1 0

1 0 0

0 1 Undef. 0 Undef.�3�33

tan �

�112

�22

�32

cos �

�1�32

�22

12

sin �

3�

2�

�

2�

3�

4�

6�

270�180�90�60�45�30�0��

333202_0404.qxd 12/7/05 11:05 AM Page 315


Using Reference Angles

Evaluate each trigonometric function.

a. b. c.

Solutiona. Because lies in Quadrant III, the reference angle is

as shown in Figure 4.44. Moreover, the cosine isnegative in Quadrant III, so

b. Because it follows that is coterminal with thesecond-quadrant angle So, the reference angle is

as shown in Figure 4.45. Finally, because the tangent isnegative in Quadrant II, you have

c. Because it follows that is coterminalwith the second-quadrant angle So, the reference angle is

as shown in Figure 4.46. Because the cosecant ispositive in Quadrant II, you have

FIGURE 4.44 FIGURE 4.45 FIGURE 4.46


=4

11

=4π

πθ

y

x′θ

= −210°

′ = 30°θ

θ

y

x

=3

4

′ =3π

π

θ

θ

y

x

� �2.

�1

sin��4�

csc 11�

4� �� csc

�

4

�� 3��4� � ��4,3��4.

11��4�11��4� � 2� � 3��4,

� ��3

3.

tan��210�� tan 30�

180� � 150� � 30�,�� 150�.

�210��210� � 360� � 150�,

� �1

2.

cos 4�

3� �� cos

�

3

�4��3� � � � ��3,�� 4��3

csc 11�

4tan��210��cos

4�

3

Example 5

Emphasize the importance of referenceangles in evaluating trigonometricfunctions of angles greater than 90�.

333202_0404.qxd 12/7/05 11:05 AM Page 316


Using Trigonometric Identities

Let be an angle in Quadrant II such that Find (a) and (b) by using trigonometric identities.

Solutiona. Using the Pythagorean identity you obtain

Substitute for

Because in Quadrant II, you can use the negative root to obtain

b. Using the trigonometric identity you obtain

Substitute for and


You can use a calculator to evaluate trigonometric functions, as shown in thenext example.

Using a Calculator

Use a calculator to evaluate each trigonometric function.

a. b. c.

SolutionFunction Mode Calculator Keystrokes Display

a. Degree 410 0.8390996

b. Radian 7

c. Radian 9 1.0641778


sec �

9

�0.6569866sin��7�cot 410�

sec �

9sin��7�cot 410�

� ��24

.

� �1

2�2

cos �.sin � tan � �1�3

�2�2�3

tan � �sin �cos �

,

� �2�2

3.

cos � � ��8�9

cos � < 0

cos2 � � 1 �1

9�

8

9.

sin �.13�1

3�2

� cos2 � � 1

sin2 � � cos2 � � 1,

tan �cos �sin � �13.�

Students often have difficulty determin-ing angles, especially when the functionsgiven are csc, sec, and/or cot. Have yourstudents rewrite the expression in termsof sin, cos, or tan, whichever is applicable,before evaluating.

TAN �� x �1 ENTER

SIN � � ENTER

COS �� x �1 ENTER

Activities

1. Determine the exact values of the sixtrigonometric functions of the anglein standard position whose terminalside contains the point

Answer:

2. For the angle find thereference angle and sketch and

in standard position.

Answer:

3. Find two values of that satisfy the equation Do not use your calculator.

Answer: � �3�

4,

7�

4

tan � � �1.�, 0 ≤ � < 2�,

θθx

= −135°′= 45°

y

�� 45�

��,

� � �135�,

tan � �73

cot � �37

cos � � �3

�58 sec � � �

�583

sin � � �7

�58 csc � � �

�587

��3, �7�.

��

Example 6

Example 7

333202_0404.qxd 12/7/05 11:05 AM Page 317


Exercises 4.4

In Exercises 1–4, determine the exact values of the sixtrigonometric functions of the angle

1. (a) (b)

2. (a) (b)

3. (a) (b)

4. (a) (b)

In Exercises 5–10, the point is on the terminal side of anangle in standard position. Determine the exact values ofthe six trigonometric functions of the angle.

5. 6.

7. 8.

9. 10.

In Exercises 11–14, state the quadrant in which lies.

11. and

12. and

13. and

14. and

In Exercises 15–24, find the values of the six trigonometricfunctions of with the given constraint.

Function Value Constraint

15. lies in Quadrant II.

16. lies in Quadrant III.

17.

18.

19.

20.

21.

22.

23. is undefined.

24. is undefined.

In Exercises 25–28, the terminal side of lies on the givenline in the specified quadrant. Find the values of the sixtrigonometric functions of by finding a point on the line.

Line Quadrant

25. II

26. III

27. III

28. IV4x � 3y � 0

2x � y � 0

y �13x

y � �x

�

�

� ≤ � ≤ 2�tan �

��2 ≤ � ≤ 3��2cot �

sec � � �1sin � � 0

sin � > 0sec � � �2

cot � < 0csc � � 4

cos � > 0cot � � �3

tan � < 0cos � �8

17

sin � < 0tan � � �158

�cos � � �45

�sin � �35

�

cot � < 0sec � > 0

tan � < 0sin � > 0

cos � > 0sin � > 0

cos � < 0sin � < 0

�

�312, �73

4��3.5, 6.8�

��5, �2��4, 10��8, 15��7, 24�

θ

(4, 4)−

y

x

(3, 1)

θ

y

x

( 4, 1)−

θ

y

x

θ

− 3, −1( )

y

x

x

( 1, 1)− θ

y

θ

y

x

(−12, −5)

x

(8, 15)−

θ

y

x

(4, 3)

θ

y

�.

VOCABULARY CHECK:

In Exercises 1– 6, let be an angle in standard position, with a point on the terminal side of and

1. 2.

3. 4.

5. 6.

7. The acute positive angle that is formed by the terminal side of the angle and the horizontal axis is called the ________ angle of and is denoted by

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

��.��

xy

� ________xr

� ________

sec � � ________tan � � ________

ry

� ________sin � � ________

r�x2 � y2 � 0.�x, y�

333202_0404.qxd 12/7/05 11:05 AM Page 318


In Exercises 29–36, evaluate the trigonometric function ofthe quadrant angle.

29. 30.

31. 32.

33. 34.

35. 36.

In Exercises 37–44, find the reference angle and sketchand in standard position.

37. 38.

39. 40.

41. 42.

43. 44.

In Exercises 45–58, evaluate the sine, cosine, and tangent ofthe angle without using a calculator.

45. 46.

47. 48.

49. 50.

51. 52.

53. 54.

55. 56.

57.

58.

In Exercises 59–64, find the indicated trigonometric valuein the specified quadrant.

Function Quadrant Trigonometric Value

59. IV

60. II

61. III

62. IV

63. I

64. III

In Exercises 65–80, use a calculator to evaluate the trigono-metric function. Round your answer to four decimal places.(Be sure the calculator is set in the correct angle mode.)

65. 66.

67. 68.

69. 70.

71. 72.

73. 74.

75. 76.

77. 78.

79. 80.

In Exercises 81–86, find two solutions of the equation. Giveyour answers in degrees and in radians

Do not use a calculator.

81. (a) (b)

82. (a) (b)

83. (a) (b)

84. (a) (b)

85. (a) (b)

86. (a) (b) sin � � ��3

2sin � �

�3

2

cot � � ��3tan � � 1

sec � � �2sec � � 2

cot � � �1csc � �2�3

3

cos � � ��2

2cos � �

�2

2

sin � � �12sin � �

12

0 ≤ � < 2�.0� ≤ � < 360�

csc��15�

14 �cot��11�

8 �sec 0.29sin��0.65�

tan��

9�tan �

9

cot 1.35tan 4.5

tan��188��sec 72�

cot 178�tan 304�

csc��330��cos��110��sec 225�sin 10�

tan �sec � � �94

sec �cos � �58

cot �csc � � �2

sec �tan � �32

sin �cot � � �3

cos �sin � � �35

�25�

4

�3�

2

10�

3

11�

4

��

2�

�

6

�

4

4�

3

�840��150�

�405�750�

300�225�

� �11�

3� � 3.5

� �7�

4� �

2�

3

� � �145�� 245�

� � 309�� 203�

��,

cot �

2csc �

cot �sin �

2

sec �sec 3�

2

csc 3�

2sin �

87. Data Analysis: Meteorology The table shows themonthly normal temperatures (in degrees Fahrenheit)for selected months for New York City andFairbanks, Alaska (Source: National ClimaticData Center)

(a) Use the regression feature of a graphing utility tofind a model of the form

for each city. Let represent the month, with corresponding to January.

t � 1t

y � a sin�bt � c� � d

�F�.�N �

Model It

Month New York Fairbanks,City, N F

January 33

April 52 32

July 77 62

October 58 24

December 38 �6

�10

333202_0404.qxd 12/7/05 11:05 AM Page 319

88. Sales A company that produces snowboards, which areseasonal products, forecasts monthly sales over the next 2 years to be

where is measured in thousands of units and is the time in months, with representing January 2006.Predict sales for each of the following months.

(a) February 2006

(b) February 2007

(c) June 2006

(d) June 2007

89. Harmonic Motion The displacement from equilibriumof an oscillating weight suspended by a spring is given by

where is the displacement (in centimeters) and is thetime (in seconds). Find the displacement when (a) (b) and (c)

90. Harmonic Motion The displacement from equilibriumof an oscillating weight suspended by a spring and subjectto the damping effect of friction is given by

where is the displacement (in centimeters) and is thetime (in seconds). Find the displacement when (a) (b) and (c)

91. Electric Circuits The current (in amperes) when 100 volts is applied to a circuit is given by

where is the time (in seconds) after the voltage is applied.Approximate the current at second after the voltageis applied.

92. Distance An airplane, flying at an altitude of 6 miles, ison a flight path that passes directly over an observer (seefigure). If is the angle of elevation from the observer tothe plane, find the distance from the observer to the planewhen (a) (b) and (c)

FIGURE FOR 92

Synthesis

True or False? In Exercises 93 and 94, determine whetherthe statement is true or false. Justify your answer.

93. In each of the four quadrants, the signs of the secantfunction and sine function will be the same.

94. To find the reference angle for an angle (given in degrees), find the integer such that

The difference is thereference angle.

95. Writing Consider an angle in standard position withcentimeters, as shown in the figure. Write a short

paragraph describing the changes in the values of and as increases continuously from

to

96. Writing Explain how reference angles are used to findthe trigonometric functions of obtuse angles.

Skills Review

In Exercises 97–106, graph the function. Identify the domainand any intercepts and asymptotes of the function.

97. 98.

99. 100.

101. 102.

103. 104.

105. 106. y � log10�x � 2�y � ln x4

y � 3 x�1 � 2y � 2x�1

h�x� �x2 � 1x � 5

f �x� �x � 7

x2 � 4x � 4

g�x� � x4 � 2x2 � 3f �x� � x3 � 8

y � 2x2 � 5xy � x2 � 3x � 4

(x, y)12 cm

θ

y

x

90�.0��tan �cos �,sin �,

x, y,r � 12

360�n � �0 ≤ 360�n � � ≤ 360�.n

�

6 mid

Not drawn to scale

θ

� � 120�.� � 90�,� � 30�,d

�

t � 0.7t

I � 5e�2t sin t

I

t �12.t �

14,

t � 0,ty

y �t� � 2e�t cos 6t

t �12.t �

14,

t � 0,ty

y�t� � 2 cos 6t

t � 1tS

S � 23.1 � 0.442t � 4.3 cos �t

6


Model It (cont inued)

(b) Use the models from part (a) to find the monthlynormal temperatures for the two cities in February,March, May, June, August, September, andNovember.

(c) Compare the models for the two cities.

333202_0404.qxd 12/7/05 11:05 AM Page 320

4.4 trigonometric functions of any angle. 312... · x 4 3 cos x r 3 5 sin y r 4 5 r x2 y2 3 2 42 25...

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