CHAPTER 7 Trigonometric Functions · SECTION 7.4 Trigonometric Functions of General Angles 541 Also, observe that if e is acute these definitions reduce to the right triangle def

540 CHAPTER 7 Trigonometric Functions

Discussion and Writing

83. Write a brief paragraph that explains how to quickly compute the trigonometric functions of 30°, 45°, and 60°.

85. Explain how you would measure the height of a TV tower that is on the roof of a tall building.

84. Explain how you would measure the width of the Grand Canyon from a point on its ridge.

Figure 37

y

Figure 38

y

OBJECTIVES 1 Find the Exact Va l ues of the Trigonometric Functions for General

Angles (p. 540)

x

DEFINITION

2 Use Coterm ina l Angles to Find the Exact Value of a Trigonometric

Function (p. 542)

3 Determine the Signs of the Trigonometric Functions of an Angle in a

Given Quad rant (p. 544)

4 Find the Reference Angle of a General Angle (p. 545)

5 Use a Reference Angle to Find the Exact Value of a Trigonometric

Function (p. 546)

6 Find the Exact Va l ues of Trigonometric Functions of an Angle, Given

I nformation a bout the Functions (p. 547)

1 Find the Exact Values of the Trigonometric Functions for General Angles

To extend the definition of the trigonometric functions to include angles that are not acute, we employ a rectangular coordinate system and place the angle in the standard position so that its vertex is at the origin and its initial side is along the positive x-axis. See Figure 37.

Let e be any angle in standard position, and let (a, b) denote the coordinates of any point, except the origin (0, 0 ) , on the terminal side of e. If

r = Va2 + b2 denotes the distance from (0, 0) to (a, b ) , then the six trigonometric functions of (J are defined as the ratios

. b sm e = r

r csc e = b

a cos e = -r

r sec e = -

a

b tan e = -

a

a cot e = b

provided no denominator equals 0. If a denominator equals 0, that trigono-metric function of the angle e is not defined. -1

Notice in the preceding definitions that if a = 0, that is, if the point (a, b) is on the y-axis, then the tangent function and the secant function are undefined. Also, if b = 0, that is, if the point (a, b) is on the x-axis, then the cosecant function and the cotangent function are undefined.

By constructing similar triangles, you should be convinced that the values of the six trigonometric functions of an angle e do not depend on the selection of the point (a, b) on the terminal side of e, but rather depend only on the angle e itself. See Figure 38 for an illustration of this when e lies in quadrant II. Since the triangles are

" 1 h . b I h . b' h' h i ' Al h . la l I SImI ar, t e ratio - equa s t e ratio -, w IC equa s sm e. so, t e ratIO - equa s r r ' r ----====\==a'=\ ===t--'-----Xx la ' i a a '

the ratio -, so - = ---; , which equals cos e. And so on. r ' r r

Figure 40

-4 -2

Figure 41 e = 0 = 0°

Figure 42 7T

-1

e = - = 90° 2

-1

E XA M P L E 1

y 2

Solution

E XA M P LE 2

y

y

Solution

p= (1 , 0)

e = 0° 1 x

p = (0, 1 )

e = 90°

x

SECTION 7.4 Trigonometric Functions of General Angles 541

Also, observe that if e is acute these definitions reduce to the right triangle definitions given in Section 7.2, as illustrated in Figure 39.

Figure 39 Y

a x

Finally, from the definition of the six trigonometric functions of a general angle, we see that the Quotient and Reciprocal Identities hold. Also, using r2 = a2 + b2 and dividing each side by r2, we can derive the Pythagorean Identities for general angles.

Finding the Exact Values of the Six Trigonometric Functions of 8, G iven a Point on the Terminal Side

Find the exact value of each of the six trigonometric functions of a positive angle e if (4, -3) is a point on its terminal side.

Figure 40 illustrates the situation. For the point (a, b) = (4, -3 ) , we have a = 4

and b = -3. Then r = Va2 + b2 = V16+9 = 5, so that

. b 3 sm e = - = - -

r 5

r 5 csc e = - = - -

b 3

a 4 cos e = - = -

r 5

r 5 sec e = - = -

a 4

'''' '::;;;,.- Now Work P R O B L E M 1 1

b 3 tan e = - = - -

a 4

a 4 cot e = - = - -

b 3 •

In the next example, we find the exact value of each of the six trigonometric 7T 37T

functions at the quadrantal angles 0, 2' 7T, and T'

Finding the Exact Values of the Six Trigonometric F unctions of Quadrantal Angles

Find the exact values of each of the six trigonometric functions of

(a) e = 0 = 0° (b) e = 7T

= 90° (c) e = 7T = 180° (d) e = 37T = 270°

2 2

(a) The point P = ( 1 , 0) is on the terminal side of e = 0 = 0° and is a distance of 1 unit from the origin. See Figure 41 . Then

(b)

sin 0 = sin 0° = Q = 0 cos 0 = cos 0° = 1:. = 1 1 1

o 1 tan 0 = tan 0° = - = 0 sec 0 = sec 0° = - = 1

1 1

Since the y-coordinate of P is 0, csc 0 and cot 0 are not defined. 7T

The point P = (0, 1 ) is on the terminal side of e = - = 90° and is a distance of 1 unit from the origin. See Figure 42. Then 2

sin 7T

= sin 90° = 1:. = 1 2 1

7T 1 csc - = csc 90° = - = 1

2 1

7T 0 cos - = cos 90° = - = 0

2 1

7T 0 cot - = cot 90° = - = 0

2 1 7T 7T .

d Since the x-coordinate of P is 0, tan - and sec - are not defme . 2 2

542 CHAPTER 7 Trigonometric F unctions

Figure 43 () = 7T = 1 800

p= (-1 , 0)

Figure 44 37T

-1

() = - = 2700 2

y

y

-1 x -1 p = (0, -1 )

x

(c) The point P = ( -1 , 0) is on the terminal side of e = 7T = 1800 and is a distance of 1 unit from the origin. See Figure 43 . Then

sin 7T = sin 1800 = Q = 0 1

o tan 7T = tan 180° = - = 0

- 1

-1 cos 7T = cos 180° = - = - 1

1 1

sec 7T = sec 180° = - = - 1 - 1

Since the y-coordinate of P is 0, CSC 7T and cot 7T are not defined.

(d) The point P = (0, - 1 ) is on the terminal side of e = 37T

= 270° and is a dis-tance of 1 unit from the origin. See Figure 44. Then 2

. 37T . 2700

- 1 1 Sll1 - = sIn = - = -

2 1

37T 1 CSC - = csc 270° = - = - 1

2 - 1

37T 0 cos - = cos 270° = - = 0

2 1

37T 0 cot - = cot 270° = - = 0

2 - 1

. 37T 37T . Since the x-coordinate of P IS 0, tan 2 and sec 2 are not defll1ed.

•

Table 4 summarizes the values of the trigonometric functions found 111 Example 2.

Table 4 (J (Radians) (J (Degrees) sin (J cos (J tan (J esc (J sec (J cot (J

0

7T

2

7T

37T

2

DEFINITION

Figure 45

00

900

1 800

2700

0 0 Not defined Not defined

0 Not defined Not defined 0

0 - 1 0 Not defined - 1 Not defined

- 1 0 Not defined - 1 Not defined 0

There is no need to memorize Table 4. To find the value of a trigonometric function of a quadrantal angle, draw the angle and apply the definition, as we did in Example 2.

2 Use Coterminal Angles to Find the Exact Value

of a Trigonometric Function

Two angles in standard position are said to be coterminal if they have the same terminal side.

See Figure 45 .

y y

x x

(a) A and B are coterminal (b) A and B are coterminal

For example, the angles 60° and 420° are coterminal, as are the angles -40° and 320°.

EXA M PL E 3

Figure 46 y

x

Figure 47

x

Figure 48 y

x 9TI "4

Figure 49 y

x

Figure 50 y

x


In general, if 0 is an angle measured in degrees, then 0 + 3600k, where k is any integer, is an angle coterminal with o. If 0 is measured in radians, then 0 + 27Tk, where k is any integer, is an angle coterminal with O.

Because coterminal angles have the same terminal side, it follows that the values of the trigonometric functions of coterminal angles are equal. We use this fact in the next example.

Using a Coterminal Angle to Find the Exact Value of a Trigonometric Function

Find the exact value of each of the following:

(a) sin 390° (b) cos 420° (c) tan 9; (d) sec( -

7;) (e) csc( -270°)

Solution (a) It is best to sketch the angle first. See Figure 46. The angle 390° is coterminal with 30°.

(b)

sin 390° = sin (30° + 360°)

. "'00 1 = sin.) = -2

See Figure 47. The angle 420° is coterminal with 60°.

cos 420° = cos( 60° + 360°)

1 = cos 60° = -2

( ) S .

8 h I 97T . . I . h

7T C ee FIgure 4 . T e ang e 4 IS cotenmna Wit "4.

97T (

7T ) tan 4 = tan "4 + 27T

7T = tan - = 1 4

( ) 77T . . .

h 7T

d See Figure 49. The angle - 4 IS cotermmal Wit "4.

( 77T) (

7T )

7T , ;::: sec -4 = sec "4 + 27T( - 1 ) = sec "4 = v2

(e) See Figure 50. The angle -270° is coterminal with 90°.

csc( -2700) = csc(90° + 360°( - 1 ) )

= csc 90° = 1 •

As Example 3 illustrates, the value of a trigonometric function of any angle is equal to the value of the same trigonometric function of an angle 0 coterminal to it, where 0° :::; 0 < 360° (or 0 :::; 0 < 27T). Because the angles 0 and 0 + 3600k (or 0 + 27Tk) , where k is any integer, are coterminal, and because the values of the trigonometric functions are equal for coterminal angles, it follows that

(J degrees

sin(O + 360°k) = sin 0 cos(O + 3600k) = cos 0 tan ( 0 + 3600k) = tan 0 csc(O + 3600k) = csc 0 sec(O + 3600k) = sec 0 cot(O + 3600k) = cot 0

where k is any integer.

(J radians

sin(O + 27Tk) = sin 0 cos(O + 27Tk) = cos 0 tan(O + 27Tk) = tan 0 csc(O + 27Tk) = csc 0 sec(O + 27Tk) = sec 0 cot(8 + 27Tk) = cot 8 (1)


3

Figure 51 () i n quadrant II, a < 0, b > 0, r > 0

y

x

Table 5

Figure 52

E XA M PL E 4

Solution

These formulas show that the values of the trigonometric functions repeat themselves every 3600 (or 27T radians) .

.... 'I!l:==;> - Now Work P R O B L E M 2 1

Determine the Signs of the Trigonometric Functions of an Angle in a Given Quadrant

If () is not a quadrantal angle, then it will lie in a particular quadrant. In such a case, the signs of the x-coordinate and y-coordinate of a point (a, b) on the terminal side

of () are known. Because r = ya2 + b2 > 0, it follows that the signs of the trigonometric functions of an angle () can be found if we know in which quadrant () lies.

For example, if () lies in quadrant II, as shown in Figure 51 , then a point (a, b) on the terminal side of () has a negative x-coordinate and a positive y-coordinate. Then,

. b SID () = - > ° r

r csc () = - > °

b

a cos () = - < ° b

tan () = - < ° r a r a

sec () = - < ° cot () = b < ° a

Table 5 lists the signs of the six trigonometric functions for each quadrant. Figure 52 provides two illustrations.

Quadrant of (J sin (J, esc (J

Positive

II Positive

III Negative

IV Negative

y II (-, +) 1(+, +) sin e > 0, csc e > ° All positive others negative

x 111(-, -) IV (+,-)

tan e > 0, cot e > 0 cos e > 0 , sec e > ° others negative others negative

(a)

cos (J, sec (J

Positive

Negative

Negative

Positive

+-+y +

x - +

+y +x + -

(b)

Finding the Quadrant in Which an Angle Lies

sine cosecant

cosine secant

tangent cotangent

tan (J, cot (J

Positive

Negative

Positive

Negative

If sin () < ° and cos () < 0, name the quadrant in which the angle () lies.

If sin () < 0, then () lies in quadrant III or IV. If cos () < 0, then () lies in quadrant II or III . Therefore, () lies in quadrant III.

•

�==,... Now Work P R O B L E M 3 3

DEFINITION

Figure 53

y

(a)

EXA M PL E 5

Solution


4 Find the Reference Angle of a General Angle

Once we know in which quadrant an angle lies, we know the sign of each trigonometric function of this angle. This information, along with the reference angle will allow us to evaluate the trigonometric functions of such an angle.

Let 8 denote an angle that lies in a quadrant. The acute angle formed by the terminal side of 8 and the x-axis is called the reference angle for 8. .J

Figure 53 illustrates the reference angle for some general angles 8. Note that a reference angle is always an acute angle. That is, a reference angle has a measure between 0° and 90°.

y y y

e

x x

(b) (c) (d)

Although formulas can be given for calculating reference angles, usually it is easier to find the reference angle for a given angle by making a quick sketch of the angle.

Finding Reference Angles

Find the reference angle for each of the following angles:

(a) 150° (b) -45° ( c) 91T

4

(a) Refer to Figure 54. The reference angle for 150° is 30°.

Figure 54 Y

x

(c) Refer to Figure 56. The refer-. 91T . 1T

ence angle for 4 IS 4' Figure 56

x

= - Now Work P R O B L E M 4 1

(d) 51T

6

(b) Refer to Figure 55. The reference angle for -45° is 45°.

Figure 55

Y x

(d) Refer to Figure 57. The refer-51T . 1T

ence angle for - 6 IS (5' Figure 57

y

x

•


THEOREM

Figure 58

y

x

EXAM P LE 6

Figure 59

x

Figure 60 y

x

Figure 61 y 1771

x

5 Use a Reference Angle to Find the Exact Value of a Trigonometric Function

The advantage of using reference angles is that, except for the correct sign, the values of the trigonometric functions of a general angle e equal the values of the trigonometric functions of its reference angle.

Reference Angles

If e is an angle that lies in a quadrant and if Cl' is its reference angle, then

sin e = ±sin Cl'

csc e = ±csc Cl'

cos e = ±cos Cl'

sec e = ±sec Cl'

tan e = ±tan Cl'

cot e = ±cot Cl'

where the + or - sign depends on the quadrant in which e lies.

(2)

.J

For example, suppose that e lies in quadrant II and Cl' is its reference angle. See

Figure 58. I f (a, b) is a point on the terminal side of e and if r = Va2 + b2, we have

and so on.

. b . S111 e = - = S111 Cl' r

a -Ial cos e = - = -- = -cos Cl'

r r

The next example illustrates how the theorem on reference angles is used.

Using the Reference Angle to Find the Exact Value of a Trigonometric Function

Find the exact value of each of the following trigonometric functions using reference angles.

(a) sin 135° (b) cos 600° 177T

(c) cos -6-

Solution (a) Refer to Figure 59. The reference angle for 135° is 45° and

. v'2 S111 45° = T. The angle 135° is in quadrant II, where the sine

function is positive, so

sin 135° = sin 450 = v'2 2

(b) Refer to Figure 60. The reference angle for 600° is 60° and cos 60° = 1:.. The angle 600° is in quadrant I II, where the cosine function is negative, so 2

cos 600° = -cos 60° = _1:. 2

. 177T 7T 7T v3 (c) Refer to Figure 61. The reference angle for -- is - and cos - = -. The

177T 6 6 6 2 angle -

6- is in quadrant II, where the cosine function is negative, so

177T 7T v3 cos-- = -cos - = ---

6 6 2

Figure 62

Figure 63

2 cos 0' = "3

-3

y

x


(d) R�er to Figure 62. The reference angle for -; is ; and tan ; = \13. The angle

- :3 is in quadrant IV, where the tangent function is negative, so

tan(-;) = -tan ; = - \13

Finding the Values of the Trigonometric Functions of a General Angle

I f the angle e i s a quadrantal angle, draw the angle, pick a point on its terminal side, and apply the definition of the trigonometric functions.

I f the angle e lies in a quadrant:

1. Find the reference angle ll' of e. 2. Find the value of the trigonometric function at ll'. 3. Adjust the sign (+ or - ) according to the sign of the trigonometric func

tion in the quadrant where e lie s.

�=Il::liI!:'- - Now Work P R O B L E M S 5 9 A N D 63

6 Find the Exact Values of Trigonometric Functions of an Angle, Given Information about the Functions

•

EXAM PLE 7 Finding the Exact Values of Trigonometric Functions

Solution

y

x

2 7T Given that cos e = - '3' '2 < e < 7T, find the exact value o f each o f the remaining

trigonometric functions.

The angle e lies in quadrant II, so we know that sin e and csc e are positive and the other four trigonometric functions are negative. I f ll' i s the reference angle for e,

2 adjacent then cos ll' = - = . TIle values o f the remaining trigonometric functions

3 hypotenuse

o f the reference angle ll' can be found by drawing the appropriate triangle and using the Pythagorean Theorem. We use Figure 63 to obtain

Vs smll' = T

3 3Vs CSCll' = -- = --

Vs 5

2 cos ll' = '3

3 sec ll' = 2:

Vs tanll' = 2

2 2Vs cotll' = -- = --

Vs 5

Now we assign the appropriate signs to each of these values to find the values of the trigonometric functions of e.

. Vs sm e = -

3-

3Vs csc e = -5-

2 cos e = - -

3

3 sec e = - -

2

1J!ln::==- Now Work P R O B L E M 8 9

Vs tan e = - --

2

2Vs cot e = ---5

•


EXAM PLE 8 Finding the Exact Values of Trigonometric Functions

If tan e = -4 and sin e < 0, find the exact value of each of the remaining trigonometric functions of e.

Solution Since tan e = -4 < ° and sin e < 0, it follows that e lies in quadrant IV. If 0' is the

Figure 64 tan a = 4

4

2 x

4 b . b . reference angle for e, then tan 0' = 4 = - = -. WIth a = 1 and = 4, we fmd

1 a r = V12 + 42 = V17. See Figure 64. Then

. 4 4V17 SInO' = -- = --V17 17

V17 cSCO' = -4-

1 V17 cOSO' = -- = --V17 17

V17 sec 0' = -- = 1

4 tan 0' = - = 4 1

1 cot 0' = "4

We assign the appropriate sign to each of these to obtain the values of the trigonometric functions of e.

-4 (1 , -4) . 4V17 Sll1e = ---17

V17 csce = ---4

V17 cose = D sece = V17

1Oi!l!>i=_"' - Now Work P R O B L E M 9 9

7.4 Assess Your Understanding

Concepts and Vocabulary

tan e = -4

1 cot e = --4

•

1. For an angle e that l ies i n quadrant III, the trigonometric functions __ and __ are pos itive.

6. True or False The reference angle is always an acute angle. 7. What is the reference angle of 6 00°?

2. Two angles in s tandard pos ition that have the s ame terminal s ide are 8. In which quadrants is the cos ine function positive?

3. T he reference angle of 2 40° is __ . 9. If 0 :5 e < 27T, for what angles e, if any, is tan e undefined?

4. True or False sin 182 ° = cos 2 °. 7T

5. True or False tan "2 is not defined.

Skill Building

o ' nIT 1 . What IS the reference angle of -3 - ?

In Problems 11-20, a point on the terminal side of an angle e is given. Find the exact value of each of the six trigonometric functions of e. "11. (-3, 4) 12. ( 5, - 12) 13. (2 , -3) 14. ( -1, -2) 15. ( -3, -3)

16. (2, -2) 17. (\13 !) 18. (_! \13) 2 '2 2 ' 2 19. (V2 _ V2)

2 ' 2

In Problems 21-32, use a coterminal angle to find the exact value of each expression. Do not use a calculator .

. 21. sin 4 05° 22. cos 42 0° 23. tan 4 05° 24. sin 3 90° 25. csc 4 500

27. cot 3 90° 28. sec 42 0° 337T

29. cos 4

In Problems 33-40, name the quadrant in which the angle e lies.

30 . 97T . SlI1 4' 31. tan(217T)

20. (_ V2 _ V2) 2 ' 2

26. sec 54 0° 97T 32. cscT

"33. sin e > 0, cos e < 0 34. s in e < 0, cos e > 0 35. s in e < 0, tan e < 0

36. cos e > 0, tan e > 0

39. sec e < 0, tan e > 0

37. cos e > 0, cot e < 0 38. s in e < 0, cot e > 0

40. csc e > 0, cot e < 0

In Problems 41-58, find the reference angle of each angle.

41. -30° 42. 60° 43. 120° 4 571'

7-. 4 271'

53. -3 48

571' . 6 771'

54 --. 6

9 871'

4 -. 3

55. 440°


44. 300°

50 771'

'4 56. 490°

45. 210°

51. -135°

1571' 57. -4

-

46. 330°

52. -240°

8 1971'

5 . -6-

In Problems 59-88, use the reference angle to find the exact value of each expression. Do not use a calculator.

59. sin 150° 60. cos 210° 61. cos 315° 62. sin 120° 63. sin 510° 64. cos 600°

65. cos( -45°)

1 . 371'

7 . SIn 4 77. sin( _

2;) 83. sin(871')

66. sine -240°) 271'

72. cos 3

78. cot( -�) 84. cos( -271')

67. sec 240° 771'

73. cot6

1471' 79. tan -3 -

85. tan(h)

68. csc 300° 771'

74. csc4

1171' 80. sec4

86. cot(571')

69. cot 330° 1371'

75. cos 4

81. csc ( -315°)

87. sec ( -371')

70. tan 225° 871'

76. tan 3

82. sec( -225°)

88. csc( _ 5;)

In Problems 89-106, find the exact value of each of the remaining trigonometric functions of 8.

89. sin 8 = ��, 8 in Quadrant I I 90. cos 8 = �, 8 i n Quadrant IV 91. cos 8 = -�, 8 in Quadrant III

92. sin 8 = - � 8 in Quadrant I I I 13' 1

95. cos 8 = - 3" 180° < 8 < 270°

1 9S. cos 8 = - 4" ' tan 8 > 0

3 101. tan 8 = 4"' sin 8 < 0

104. sec e = -2, tan e > 0

93. sin 8 = :3' 90° < 8 < 180°

96. sin 8 = - � , 180° < 8 < 270°

99. sec 8 = 2, sin 8 < 0

4 102. cot e = 3" cos f) < 0

105. csc 8 = -2, tan e > 0

107. Find the exact value of sin 40° + sin 130° + sin 220° + sin 310°.

lOS. Find the exact value of tan 40° + tan 140°.

Applications and Extensions

4 94. cos e = 5' 270° < e < 360°

. 2 0 97. SIn e = 3" tan e <

100. csc e = 3, cot e < 0

1 103. tan 8 = - 3' ' sin e > 0

106. cot 8 = -2, sec 8 > 0

109. If f (8) = sin 8 = 0.2, find f(8 + 71') . 110. If gee) = cos 8 = 0.4, find g(8 + 71') . 111. If F (8) = tan e = 3, find F(e + 71') .

112. IfG (e) = cot 8 = -2, find G (8 + 71'). 113. If sin e = � , f indCSC(8 + 71'). 2 . 114. If cos 8 = 3' , fmd sec (8 + 71').

115. Find the exact value of sin 1° + sin 2° + sin 3° + . . . + sin 358° + sin 359°

116. F ind the exact value of cos 1 ° + cos 2° + cos 3° + . . . + cos 358° + cos 359°

117. Projectile Motion An object is propelled upward at an angle 8, 45° < 8 < 90° , to the horizontal with an initial velocity of Vo feet per second fro m the base of a plane that makes an angle of 45° with the horizontal. See the i llustration. If air resistance is ignored, the distance R that it travels up the inclined plane is given by the function

v2yz R (8) = � [sin(28) - cos(28) - 1J

Discussion and Writing

118. Give three examples that demonstrate how to use the theorem on reference angles.

(a) Find the distance R that the object travels along the inclined plane if the initial velocity is 32 feet per second and 8 = 60°.

b;l (b) Graph R = R( e ) if the initial velocity is 32 feet per second.

(c) What value of 8 makes R largest?

119. Write a brief paragraph that explains how to quickly compute the value of the trigonometric functions of 0° , 90°, 180°, and 270°.

Documents

CHAPTER 7 Trigonometric Functions · SECTION 7.4 Trigonometric Functions of General Angles 541 Also, observe that if e is acute these definitions reduce to the right triangle def