crunchy math · CHAPTER 4 Trigonometric Functions Section 4.1 Radian and Degree Measure . . . . . . . . . . . . . . . . 272 Section 4.2 Trigonometric Functions: The Unit Circle

C H A P T E R 4Trigonometric Functions

Section 4.1 Radian and Degree Measure . . . . . . . . . . . . . . . . 272

Section 4.2 Trigonometric Functions: The Unit Circle . . . . . . . . 281

Section 4.3 Right Triangle Trigonometry . . . . . . . . . . . . . . . . 289

Section 4.4 Trigonometric Functions of Any Angle . . . . . . . . . . 300

Section 4.5 Graphs of Sine and Cosine Functions . . . . . . . . . . . 317

Section 4.6 Graphs of Other Trigonometric Functions . . . . . . . . . 329

Section 4.7 Inverse Trigonometric Functions . . . . . . . . . . . . . . 339

Section 4.8 Applications and Models . . . . . . . . . . . . . . . . . . 350

Review Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 360

Practice Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 377

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C H A P T E R 4Trigonometric Functions

Section 4.1 Radian and Degree Measure

272

You should know the following basic facts about angles, their measurement, and their applications.

! Types of Angles:

(a) Acute: Measure between and

(b) Right: Measure

(c) Obtuse: Measure between and

(d) Straight: Measure

! and are complementary if They are supplementary if

! Two angles in standard position that have the same terminal side are called coterminal angles.

! To convert degrees to radians, use radians.

! To convert radians to degrees, use 1 radian

! one minute of

! one second of

! The length of a circular arc is where is measured in radians.

! Speed

! Angular speed ! "!t ! s!rt

! distance!time

"s ! r"

1#! 1!60 of 1$ ! 1!36001% !

1#! 1!601$ !

! "180!&##.1# ! &!180

' ( ) ! 180#.' ( ) ! 90#.)'

180#.

180#.90#

90#.

90#.0#

1. The angle shown is approximately 2 radians.

Vocabulary Check1. Trigonometry 2. angle 3. standard position 4. coterminal

5. radian 6. complementary 7. supplementary 8. degree

9. linear 10. angular

2. The angle shown isapproximately radians.*4

3. (a) Since lies in Quadrant IV.

(b) Since lies in

Quadrant II.

5&2

<11&

4< 3&,

11&4

3&2

<7&4

< 2&, 7&4

4. (a) Since lies in

Quadrant IV.

(b) Since lies in

Quadrant II.

*3&2

< *13&

9< *&,

13&9

*&2

< *5&12

< 0, 5&2

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Section 4.1 Radian and Degree Measure 273

9. (a)

(b)

x

y

!3

2

2&3

x

116!

y11&6

7. (a)

(b)

x

y

!3

4

4&3

43!

x

y13&4

5. (a) Since lies in Quadrant IV.

(b) Since lies in

Quadrant III.

*& < *2 < *&

2; *2

*&

2< *1 < 0; *1 6. (a) Since 3.5 lies in Quadrant III.

(b) Since 2.25 lies in Quadrant II.&

2< 2.25 < &,

& < 3.5 <3&

2,

8. (a)

(b)

x

y

52!"

*5&2

x

74" !

y

*7&4

10. (a) 4

(b)

x

y

"3

*3

x

4

y

11. (a) Coterminal angles for

&

6* 2& ! *

11&

6

&

6( 2& !

13&

6

&

6: (b) Coterminal angles for

2&

3* 2& ! *

4&

3

2&

3( 2& !

8&

3

2&

3:

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274 Chapter 4 Trigonometric Functions


(b) Coterminal angles for

5&

4* 2& ! *

3&

4

5&

4( 2& !

13&

4

5&4

:

7&

6* 2& ! *

5&

6

7&

6( 2& !

19&

6

7&6

: 13. (a) Coterminal angles for


*2&

15* 2& ! *

32&

15

*2&

15( 2& !

28&

15

*2&15

:

*9&

4( 4& !

7&

4

*9&

4( 2& ! *

&

4

*9&4

: 14. (a) Coterminal angles for


8&

45* 2& ! *

82&

45

8&

45( 2& !

98&

45

8&45

:

7&

8* 2& ! *

9&

8

7&

8( 2& !

23&

8

7&8

:

15. Complement:

Supplement: & *&3

!2&3

&2

*&3

!&6

17. Complement:

Supplement: & *&6

!5&6

&2

*&6

!&3

16. Complement: Not possible; is greater than

(a)Supplement: & *3&

4!

&

4

&

2.

3&

4

18. Complement: Not possible; is greater than

Supplement: & *2&3

!&3

&

2.

2&

3

19. Complement:

Supplement: & * 1 $ 2.14

&

2* 1 $ 0.57 20. Complement: None

Supplement: & * 2 $ 1.14

%2 >&2&

21.

The angle shown is approximately 210#.

22.

The angle shown is approximately *45#.

23. (a) Since lies in Quadrant II.

(b) Since lies inQuadrant IV.

282#270# < 282# < 360#,

150#90# < 150# < 180#,

25. (a) Since lies in Quadrant III.

(b) Since lies in Quadrant I.*336# 30$

*360# < *336# 30$ < *270#,

*132# 50$*180# < *132# 50$ < *90#,

24. (a) Since lies in Quadrant I.

(b) Since lies in Quadrant I.8.5#0# < 8.5# < 90#,

87.9#0# < 87.9# < 90#,

26. (a) Since lies in Quadrant II.

(b) Since liesin Quadrant IV.

*12.35#*90# < *12.35# < *0#,

*245.25#*270# < *245.25# < *180#,

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27. (a)

30!x

y30# (b)

150°

x

y150#

28. (a)

(b)

x

"120°

y

*120#

x

"270°

y

*270# 29. (a)

(b)

780°

x

y

780#

x

405°

y

405# 30. (a)

(b)

x

y

"600!

*600#

! °450

x

y

*450#



*36# * 360# ! *396#

*36# ( 360# ! 324#

*36#:

52# * 360# ! *308#

52# ( 360# ! 412#

52#: 33. (a) Coterminal angles for


230# * 360# ! *130#

230# ( 360# ! 590#

230#:

300# * 360# ! *60#

300# ( 360# ! 660#300#:32. (a) Coterminal angles for


*390# ( 360# ! *30#

*390# ( 720# ! 330#

*390#:

114# * 360# ! *246#

114# ( 360# ! 474#

114#:



*740# ( 720# ! *20#

*740# ( 1080# ! 340#

*740#:

*445# ( 360# ! *85#

*445# ( 720# ! 275#

*445#: 35. Complement:

Supplement: 180# * 24# ! 156#

90# * 24# ! 66# 36. Complement: Not possible

Supplement: 180# * 129# ! 51#

37. Complement:

Supplement: 180# * 87# ! 93#

90# * 87# ! 3# 38. Complement: Not possible

Supplement: 180# * 167# ! 13#

39. (a)

(b) 150# ! 150#% &

180#& !5&

6

30# ! 30#% &

180#& !&

6

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44. (a)

(b) 3& ! 3&%180#& & ! 540#

*4& ! *4&%180#& & ! *720#

46. (a)

(b)28&15

!28&15 %180#

& & ! 336#

*15&

6! *

15&6 %180#

& & ! *450#45. (a)

(b) *13&60

! *13&60 %180#

& & ! *39#

7&3

!7&3 %180#

& & ! 420#

47. 115# ! 115% &

180#& $ 2.007 radians 48. radians83.7# ! 83.7#% &180#& $ 1.461

50. radians*46.52# ! *46.52#% &180#& $ *0.81249. *216.35# ! *216.35% &

180#& $ *3.776 radians

51. *0.78# ! *0.78% &180#& $ *0.014 radians

53.&

7!

&

7%180#

& & $ 25.714# 55. 6.5& ! 6.5&%180#& & ! 1170#

61. 85# 18$ 30% ! 85# ( "1860## ( " 30

3600## $ 85.308#

63. *125# 36% ! *125# * " 363600## ! *125.01#

65. 280.6# ! 280# ( 0.6"60#$ ! 280# 36$

67. *345.12# ! *345# 7$ 12%

57. *2 ! *2%180#

& & $ *114.592#

59. 64# 45$ ! 64# ( %4560&#

! 64.75#

52. radians395# ! 395#% &180#& $ 6.894

54.8&13

!8&13%180#

& & $ 110.769#

56. *4.2& ! *4.2&%180#

& & ! *756#

58. *0.48 ! *0.48%180#& & $ *27.502#

60. *124# 30$ ! *124.5#

62. *408# 16$ 25% $ *408.274#

64. 330# 25% $ 330.007#

66. *115.8# ! *115# 48$ 68. 310.75# ! 310# 45$

43. (a)

(b) *7&

6! *

7&

6 %180#

& & ! *210#

3&

2!

3&

2 %180#

& & ! 270#

40. (a)

(b) 120# ! 120#% &

180#& !2&

3

315# ! 315#% &

180#& !7&

4

42. (a) (b) 144# ! 144#% &

180#& !4&

5*270# ! *270#% &

180#& ! *3&

2

41. (a)

(b) *240# ! *240#% &

180#& ! *4&

3

*20# ! *20#% &

180#& ! *&

9

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69.

$ *20.34# ! *20# 20$ 24%

*0.355 ! *0.355%180#& & 70.

$ 45# 3$ 47%

! 45# ( 3$ ( 0.786"60% #

! 45# ( "0.0631#"60$#

$ 45.0631#

0.7865 ! 0.7865%180#

& &

72.

radians " ! 3112 ! 2 7

12

31 ! 12"

s ! r"71.

radians" ! 65

6 ! 5"

s ! r" 73.

radians3" ! 327 ! 44

7

32 ! 7"

3s ! r"

74.

Because the angle represented is clockwise, this angle is radians.*45

" ! 6075 ! 4

5 radians

60 ! 75"

s ! r"

78. radian" !sr

!1022

!511

75. The angles in radians are:

90# !&2

60# !&3

45# !&4

30# !&6

0# ! 0

330# !11&

6

270# !3&2

210# !7&6

180# ! &

135# !3&4

76. The angles in degrees are:

5&6

! 150#

2&3

! 120#

&3

! 60#

&4

! 45#

&6

! 30#

& ! 180#

7&4

! 315#

5&3

! 300#

4&3

! 240#

5&4

! 225#

77.

radians " !815

8 ! 15"

s ! r" 79.

radians " !7029

$ 2.414

35 ! 14.5"

s ! r"

80. kilometers, kilometers

" !sr

!16080

! 2 radians

s ! 160r ! 80

82. feet,

s ! r" ! 9%&

3& ! 3& feet

" ! 60# !&

3r ! 9

81. in radians

inches s ! 14"180#% &180& ! 14& $ 43.982

s ! r", "

83. in radians

meters meters$ 56.55 s ! 27%2&3 & ! 18&

s ! r", "

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90.

s ! r" ! 4000"1.0105# $ 4042.0 miles

" ! 31# 46$ ( 26# 8$ ! 57# 54$ $ 1.0105 rad

r ! 4000 miles

89.

s ! r" ! 4000"0.28537# $ 1141.48 miles

! 16# 21$ 1% $ 0.28537 radian

" ! 42# 7$ 33% * 25# 46$ 32%88. r !s"

!8

330#"&!180## !48

11& inches $ 1.39 inches

91.

$ 4# 2$ 33.02%

" !sr

!4506378

$ 0.07056 radian $ 4.04#

93. $ 23.87#" !sr

!2.56

!2560

!512

radian

95. (a) single axel:

(b) double axel:

(c) triple axel: ! 7& radians 312 revolutions ! 1260#

! 4& ( & ! 5& radians

212 revolutions ! 720# ( 180# ! 900#

! 2& ( & ! 3& radians

112 revolutions ! 360# ( 180# ! 540#

96. Linear speed !st

!r"t

!"6400 ( 1250#2&

110$ 436.967 km!min

92.

" !sr

!4006378

$ 0.0627 rad $ 3.59#

r ! 6378, s ! 400

94. " !sr

!245

! 4.8 rad $ 275.02#

97. (a)

Angular speed

(b) Radius of saw blade

Radius in feet

Speed

! 0.3125"80&# ! 78.54 ft!sec

!st

!r"

t! r

"

t! r"angular speed#

!3.7512

! 0.3125 ft

!7.52

! 3.75 in.

! "2&#"40# ! 80& rad!sec

RevolutionsSecond

!240060

! 40 rev!sec 98. (a)

Angular speed

(b) Radius of saw blade

Radius in feet

Speed

! 0.3021"160&# $ 151.84 ft!sec

!st

!r"

t! r

"

t! r"angular speed#

!3.62512

$ 0.3021 ft

!7.25

2! 3.625 in.

! "2&#"80# ! 160& rad!sec

RevolutionsSecond

!480060

! 80 rev!sec

86. r !s"

!3

4&!3!

94&

meters $ 0.72 meter 87. r !s"

!82

135#"&!180## !3283&

miles $ 34.80 miles

84. centimeters,

centimeters cm$ 28.27s ! r" ! 12%3&4 & ! 9&

" !3&

4r ! 12 85. r !

s"

!36

&!2!

72&

feet $ 22.92 feet

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99. (a)

Angular speed

Radius of wheel

Speed

(b) Let spin balance machine rate.

70 ! r"angular speed# ! r%2& +x

"1!60#& !5

25,344120&x " x $ 941.18 rev!min

x !

!5

25,344 + 57,600& !125&

11$ 35.70 miles!hr

!st

!r"

t! r

"

t! r"angular speed#

!25!2

"12 in.!ft#"5280 ft!mi# !5

25,344 miles

! 2& "28,800# ! 57,600& rad!hr

RevolutionsHour

!480

"1!60# ! 28,800 rev!hr

100. (a)

(b) Speed

For the outermost track, 6000& cm!min

6"400&# # linear speed # 6"1000&#

!st

!r"

t! r

"

t! r"angular speed# ! 6"angular speed#

400& rad!min # angular speed # 1000& rad!min

2&"200# # angular speed # 2&"500#

200 #revolutions

minute# 500

101. False, 1 radian so one radian

is much larger than one degree.

! %180& &# $ 57.3#,

103. True:2&3

(&4

(&12

!8& ( 3& ( &

12! & ! 180#

102. No, is coterminal with 180#.*1260#

104. (a) An angle is in standard position when the origin is the vertex and the initial sidecoincides with the positive -axis.

(b) A negative angle is generated by a clockwise rotation.

(c) Angles that have the same initial and terminal sides are coterminal angles.

(d) An obtuse angle is between and 180#.90#

x

105. If is constant, the length of the arc isproportional to the radius and hence increasing.

"s ! r"#," 106. Let A be the area of a circular sector of radius

and central angle Then

A&r2 !

"

2& " , !

12

r2".

".r

107. square metersA !12

r2" !12

"10#2 +&

3!

503

& 108. Because

Hence, A !12

r2" !12

152%1215& ! 90 ft2.

s ! r", " !1215

.

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109.

(a) Domain:

Domain:

The area function changes more rapidly for because it is quadratic and the arc length function is linear.

(b) Domain:

Domain: 0 < " < 2& s ! r" ! 10"

00

2

320

A

s

"

0 < " < 2& r ! 10 " A ! 12"102#" ! 50"

r > 1

r > 0 s ! r" ! r"0.8#

r > 0 " ! 0.8 " A ! 12r2"0.8# ! 0.4r2

00

12

8

As

A ! 12r2", s ! r"

113.

"2 2 3 4 5 6

"2

"3

"4

1

2

3

4

x

y

115.

x

y

"2"3"4 1 2 3 4

"3

1

3

4

5

117.

x

y

"2"3"4 1 2 3 4

"2

"4

"5

1

2

3

110. If a fan of greater diameter is installed, the angular speed does not change.

111. Answers will vary. 112. Answers will vary.

114.

x

y

"2"3"4 1 2 3 4

"2

"3

"6

1

2

116.

x

y

"3"4"5 1 2 3

"2

"3

"4

1

2

3

4

118.

x

y

"2 1 2 3 5 6

"2

"3

"4

1

2

3

4

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Section 4.2 Trigonometric Functions: The Unit Circle

Section 4.2 Trigonometric Functions: The Unit Circle 281

2.

cot ! "xy

"12!135!13

"125

sec ! "1x

"1

12!13"

1312

csc ! "1y

"1

5!13"

135

tan ! "yx

"5!1312!13

"512

cos ! " x "1213

sin ! " y "513

"x, y# " $1213

, 513%

4.

cot ! "xy

"#4!5#3!5

"43

sec ! "1x

"1

#4!5" #

54

csc ! "1y

"1

#3!5" #

53

tan ! "yx

"#3!5#4!5

"34

cos ! " x " #45

sin ! " y " #35

"x, y# " $#45

, #35%

6. t "$

3 ! $1

2, &32 %

1.

csc ! "1y

"1715

sec ! "1x

" #178

cot ! "xy

" #815

tan ! "yx

" #158

cos ! " x " #817

sin ! " y "1517

3.

csc ! "1y

" #135

sec ! "1x

"1312

cot ! "xy

" #125

tan ! "yx

" #512

cos ! " x "1213

sin ! " y " #513

Vocabulary Check1. unit circle 2. periodic 3. odd, even

5. corresponds to $&22

, &22 %.t "

$

4

! You should know how to evaluate trigonometric functions using the unit circle.

! You should know the definition of a periodic function.

! You should be able to recognize even and odd trigonometric functions.

! You should be able to evaluate trigonometric functions with a calculator in both radian and degree mode.

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10. corresponds to $12

, #&32 %.t "

5$3

12. t " $ ! "#1, 0#

18. corresponds to

tan $

3"

yx

"&3!21!2

" &3

cos $

3" x "

12

sin $

3" y "

&32

$12

, &32 %.t "

$

3

9. corresponds to $#12

, &32 %.t "

2$3

11. corresponds to "0, #1#.t "3$

2

13. corresponds to $&22

, &22 %.t " #

7$

414. corresponds to $#

12

, &32 %.t " #

4$

3

15. corresponds to "0, 1#.t " #3$

216. corresponds to "1, 0#.t " #2$

17. corresponds to

tan t "yx

" 1

cos t " x "&22

sin t " y "&22

$&22

, &22 %.t "

$

4

19. corresponds to

tan t "yx

"1&3

"&33

cos t " x " #&32

sin t " y " #12

$#&32

, #12%.t "

7$

620. corresponds to

tan t "yx

" #1

cos t " x " #&22

sin t " y "&22

$#&22

, &2

2 %.t " #5$

4

21. corresponds to

tan t "yx

" #&3

cos t " x " #12

sin t " y "&3

2

$#12

, &32 %.t "

2$

322. corresponds to

tan t "yx

" #&3

cos t " x "12

sin t " y " #&3

2

$12

, #&32 %.t "

5$

3

8. t "5$

4 ! $#

&22

, #&22 %7. corresponds to $#

&32

, #12%.t "

7$6

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24. corresponds to

tan t "yx

" #&33

cos t " x "&32

sin t " y " #12

$&32

, #12%.t "

11$

623. corresponds to

tan t "yx

" &3

cos t " x "12

sin t " y "&32

$12

, &32 %.t " #

5$3

25. corresponds to

tan t "yx

" #&33

cos t " x "&32

sin t " y " #12

$&32

, #12%.t " #

$

626. corresponds to

tan$#3$4 % "

yx

" 1

cos$#3$4 % " x " #

&22

sin$#3$4 % " y " #

&22

$#&22

, #&22 %.t " #

3$

4

27. corresponds to

tan t "yx

" 1

cos t " x "&22

sin t " y "&22

$&22

, &22 %.t " #

7$4

28. corresponds to

tan t "yx

" #&3

cos t " x " #12

sin t " y "&32

$#12

, &32 %.t " #

4$3

29. corresponds to

is undefined.tan t "yx

cos t " x " 0

sin t " y " 1

"0, 1#.t " #3$

2

31. corresponds to

cot t "xy " #1tan t "

yx " #1

sec t "1x

" #&2cos t " x " #&22

csc t "1y

" &2sin t " y "&22

$#&22

, &22 % .t "

3$

4

30. corresponds to

tan"#2$# "yx

"01

" 0

cos"#2$# " x " 1

sin"#2$# " y " 0

"1, 0#.t " #2$

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.


32. corresponds to

cot t "xy

" #&3

sec t "1x

" #2&3

" #2&3

3

csc t "1y

" 2

tan t "yx

" #1&3

" #&33

cos t " x " #&32

sin t " y "12

$#&32

, 12%.t "

5$6

33. corresponds to

is undefined.

is undefined. cot t "xy " 0tan t "

yx

sec t "1x

cos t " x " 0

csc t "1y

" 1sin t " y " 1

"0, 1#.t "$

2

35. corresponds to

cot ! "&33

tan t "yx " &3

sec ! " #2cos t " x " #12

csc t " #2&3

3sin t " y " #

&32

$#12

, #&32 %.t " #

2$

334. corresponds to

undefined

undefined

cot 3$

2"

xy

"0

#1" 0

sec 3$

2"

1x

"10

!

csc 3$

2"

1y

"1

#1" #1

tan 3$

2"

yx

"#10

!

cos 3$

2" x " 0

sin 3$

2" y " #1

"0, #1#.t "3$

2

36. corresponds to

tan$#7$4 % "

yx

" 1

cos$#7$4 % " x "

&22

sin$#7$4 % " y "

&22

$&22

, &22 %.t " #

7$

4

cot$#7$4 % "

xy

" 1

sec$#7$4 % "

1x

" &2

csc$#7$4 % "

1y

" &2

37. sin 5$ " sin $ " 0©

Hou

ghto

n M

ifflin

Com

pany

. All

right

s re

serv

ed.


39. cos 8$

3" cos

2$

3" #

12

38. Because

cos 7$ " cos"6$ % $# " cos $ " #1

7$ " 6$ % $ :

40. Because

sin 9$

4" sin$2$ %

$

4% " sin $

4"

&22

9$

4" 2$ %

$

4:

42. Because

" sin$5$6 % "

12

sin$#19$

6 % " sin$#4$ %5$6 %

#19$

6" #4$ %

5$6

:

44. Because :

" cos 4$

3" #

12

cos$#8$

3 % " cos$#4$ %4$

3 %

#8$

3" #4$ %

4$

3

41. cos$#13$

6 % " cos$#$6% " cos$11$

6 % "&32

43. sin$#9$

4 % " sin$#$

4% " #&22

45.

(a)

(b) csc"#t# " #csc t " #3

sin"#t# " #sin t " #13

sin t "13

47.

(a)

(b) sec"#t# "1

cos"#t# " #5

cos t " cos"#t# " #15

cos"#t# " #15

46. cos

(a)

(b) sec"#t# "1

cos"#t# "1

cos t" #

43

cos"#t# " cos t " #34

t " #34

48.

(a)

(b) csc t "1

sin"t# "1

#sin"#t# " #83

sin t " #sin"#t# " #38

sin"#t# "38

49.

(a)

(b) sin"t % $# " #sin t " #45

sin"$ # t# " sin t "45

sin t "45

50.

(a)

(b) cos"t % $# " #cos t " #45

cos"$ # t# " #cos t " #45

cos t "45

51. sin 7$9

' 0.6428

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56. cot 3.7 "1

tan 3.7' 1.6007

52. tan 2$5

' 3.0777 53. cos 11$

5' 0.8090 54. sin

11$9

' #0.6428

55. csc 1.3 ' 1.0378 57. cos"#1.7# ' #0.1288

59. csc 0.8 "1

sin 0.8' 1.3940

76.

(a)

(b)

y"0# "14

e#0 cos"0# "14

foot

y"t# "14

e#t cos 6t

70. (a)

(b) cos 2.5 " x ' #0.8

sin 0.75 " y ' 0.7

72. (a)

(b)

t ' 0.72 or t ' 5.56

cos t " 0.75

t ' 4.0 or t ' 5.4

sin t " #0.75

58. cos"#2.5# ' #0.8011 60. sec 1.8 "1

cos 1.8' #4.4014

62. sin"#13.4# ' #0.740461. sec 22.8 "1

cos 22.8' #1.4486

69. (a)

(b) cos 2 ' #0.4

sin 5 ' #1

63. cot 2.5 "1

tan 2.5' #1.3386

64. tan 1.75 ' #5.5204 65. csc"#1.5# "1

sin"#1.5# ' #1.0025

66. tan"#2.25# ' 1.2386 67. sec"#4.5# "1

cos"#4.5# ' #4.7439

68. csc"#5.2# "1

sin"#5.2# ' 1.1319

71. (a)

(b)

t ' 1.82 or 4.46

cos t " #0.25

t ' 0.25 or 2.89

sin t " 0.25 73.

amperes ' 0.79

I"0.7# " 5e#1.4 sin 0.7

I " 5e#2t sin t

74. At

I ' 5e#2"1.4# sin 1.4 ' 0.30 amperes.

t " 1.4, 75.

(a)

(b)

(c) y"12# " 1

4 cos 3 ' #0.2475 ft

y"14# " 1

4 cos 32 ' 0.0177 ft

y"0# " 14 cos 0 " 0.2500 ft

y"t# " 14 cos 6t

(c) The maximum displacements are decreasingbecause of friction, which is modeled by the

term.

(d)

t "$12

, $4

6t "$2

, 3$2

cos 6t " 0

14

e#t cos 6t " 0

e#t

0.50 1.02 1.54 2.07 2.59

0.09 0.03 #0.02#0.05#0.15y

t

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78. True

80. True

82. (a) The points and are symmetric about the origin.

(b) Because of the symmetry of the points, you canmake the conjecture that

(c) Because of the symmetry of the points, you canmake the conjecture that cos"t1 % $# " #cos t1.

sin"t1 % $# " #sin t1.

"x2, y2#"x1, y1#

84.

Therefore, sin t1 % sin t2 & sin"t1 % t2#.

sin 1 ' 0.8415

sin"0.25# % sin"0.75# ' 0.2474 % 0.6816 " 0.9290

77. False. sin$#4$3 % "

&32

> 0

79. False. 0 corresponds to "1, 0#.

81. (a) The points have -axis symmetry.

(b) since they have the same-value.

(c) since the -values haveopposite signs.

x#cos t1 " cos "$ # t1#

ysin t1 " sin "$ # t1#

y

83.

Thus, cos 2t & 2 cos t.

2 cos 0.75 ' 1.4634cos 1.5 ' 0.0707,

85.

!!"

(x, y)

(x, "y)

x

y

sec ! "1

cos !"

1cos"#!# " sec"#!#

cos ! " x " cos"#!#

87. is odd.

h"#t# " f "#t#g"#t# " #f "t#g"t# " #h"t#

h"t# " f "t#g"t# 88. and

Both and are odd functions.

The function is even.h"t# " f"t#g"t#

" sin t tan t " h"t#

" "#sin t#"#tan t#

h"#t# " sin"#t# tan"#t#

h"t# " f"t#g"t# " sin t tan t

gf

g"t# " tan tf"t# " sin t

86.

cot ! "xy

" #cot"#!#

tan ! "yx

" #tan"#!#

csc ! "1y

" #csc"#!#

sin ! " y " #sin"#!#

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90.

f#1"x# " 3&4"x # 1#

y " 3&4"x # 1#

4"x # 1# " y3

x # 1 " 14 y3

x " 14 y3 % 1

y " 14x3 % 1

"9 9

"6

6

f f"1

f"x# " 14 x3 % 189.

f #1"x# " 23"x % 1#

23"x % 1# " y

2x % 2 " 3y

2x " 3y # 2

x " 12"3y # 2#

y " 12"3x # 2#

"9 9

"6

6

ff"1

f "x# " 12"3x # 2#

92.

f#1"x# "x

2 # x, x < 2

x

2 # x" y, x < 2

x " y"2 # x#

x " 2y # xy

xy % x " 2y

x "2y

y % 1

y "2x

x % 1, x > #1

"3 9

"4

4 f"x# "

2xx % 1

, x > #191.

f#1"x# " &x2 % 4, x " 0

&x2 % 4 " y, x " 0

x2 % 4 " y2

x2 " y2 # 4

x " &y2 # 4

"2 10

"1

7

ff"1

y " &x2 # 4

f "x# " &x2 # 4, x " 2, y " 0

94.

Asymptotes:

10864

468

10

"6"8

x

y

x " #3, x " 2, y " 0

f "x# "5x

x2 % x # 6"

5x"x % 3#"x # 2#93.

Asymptotes:

2 4 5 6 7 8"2 "1

1

345678

"2"3"4

"3"4x

y

x " 3, y " 2

f "x# "2x

x # 3

95.

Asymptotes: x " #2, y "12

x & 2 "x % 5

2"x % 2#,

f "x# "x2 % 3x # 10

2x2 # 8"

"x # 2#"x % 5#2"x # 2#"x % 2#

x

y

"1"5"6"7 1"1

"2

"3

"4

1

2

3

4

"2

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Section 4.3 Right Triangle Trigonometry 289

96.

Slant asymptote:

Vertical asymptotes: x ! 3.608, x ! !1.108

y "x2

!74 4

68

10

!8!10

!2!4!6!8 6 8 10

2

y

x

f "x# "x3 ! 6x2 # x ! 1

2x2 ! 5x ! 8"

x2

!74

!15"x # 4#

4"2x2 ! 5x ! 8#

97.

Domain: all real numbers

Intercepts:

No asymptotes

"0, !4#, "1, 0#, "!4, 0#

0 " x2 # 3x ! 4 " "x # 4#"x ! 1# ! x " 1, !4

y " x2 # 3x ! 4 98.

Domain: all

Intercepts:

Asymptote: x " 0

"1, 0#, "!1, 0#

ln x4 " 0 ! x4 " 1 ! x " ±1

x $ 0

y " ln x4

99.


Intercept:

Asymptote: y " 2

"0, 5#

f"x# " 3x#1 # 2 100.


Intercepts:

Asymptotes: x " !2, y " 0

"7, 0#, $0, !74%

x $ !2

f"x# "x ! 7

"x2 # 4x # 4# "x ! 7

"x # 2#2

Section 4.3 Right Triangle Trigonometry

! You should know the right triangle definition of trigonometric functions.

(a) (b) (c)

(d) (e) (f)

! You should know the following identities.

(a) (b) (c)

(d) (e) (f)

(g) (h) (i)

( j) (k)

! You should know that two acute angles are complementary if and cofunctions of complementary angles are equal.

! You should know the trigonometric function values of and or be able to construct triangles fromwhich you can determine them.

60%,30%, 45%,

& # ' " 90%,& and '

1 # cot2 ( " csc2 (1 # tan2 ( " sec2 (

sin2 ( # cos2 ( " 1cot ( "cos (sin (

tan ( "sin (cos (

cot ( "1

tan (tan ( "

1cot (

sec ( "1

cos (

cos ( "1

sec (csc ( "

1sin (

sin ( "1

csc (

cot ( "adjopp

sec ( "hypadj

csc ( "hypopp

Adjacent side

Opp

osite

sid

eHyp

otenu

se

"

tan ( "oppadj

cos ( "adjhyp

sin ( "opphyp

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Vocabulary Check1. (a) iii (b) vi (c) ii (d) v (e) i (f) iv

2. hypotenuse, opposite, adjacent 3. elevation, depression

1.

cot ( "adjopp

"43

sec ( "hypadj

"54

csc ( "hypopp

"53

tan ( "oppadj

"34

cos ( "adjhyp

"45

sin ( "opphyp

"35

adj " &52 ! 32 " &16 " 4

35

"

3.

cot ( "adjopp

"158

sec ( "hypadj

"1715

csc ( "hypopp

"178

tan ( "oppadj

"815

cos ( "adjhyp

"1517

sin ( "opphyp

"817

hyp " &82 # 152 " 17

"15

8

2.

sin

cos

tan

csc

sec

cot ( "adjopp

"125

( "hypadj

"1312

( "hypopp

"135

( "oppadj

"512

( "adjhyp

"1213

( "opphyp

"513

b " &132 ! 52 " &169 ! 25 " 12

513

b"

4.

sin

cos

tan

csc

sec

cot ( "23

( "&13

2

( "&13

3

( "1812

"32

( "12

6&13"

2&1313

( "18

6&13"

3&1313

C " &182 # 122 " &468 " 6&13

12

18 c

"

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6.

sin

cos

tan

csc

sec

cot ( "adjopp

"&161

8

( "hypadj

"15

&161"

15&161161

( "hypopp

"158

( "oppadj

"8

&161"

8&161161

( "adjhyp

"&161

15

( "opphyp

"815

15 8

"

adj " &152 ! 82 " &161

sin

cos

tan

csc

sec

cot ( "adjopp

"&1612 ) 4

"&161

8

( "hyp

adj"

7.5

"&161'2# "15

&161"

15&161

161

( "hypopp

"7.54

"158

( "opp

adj"

4

"&161'2# "8

&161"

8&161

161

( "adjhyp

"&1612 ) 7.5

"&161

15

( "opphyp

"4

7.5"

815

7.5 4"

adj " &7.52 ! 42 "&161

2

The function values are the same because the triangles are similar, and corresponding sides are proportional.

5.

cot ( "adjopp

"86

"43

sec ( "hypadj

"108

"54

csc ( "hypopp

"106

"53

tan ( "oppadj

"68

"34

cos ( "adjhyp

"810

"45

sin ( "opphyp

"610

"35

"8

10

opp " &102 ! 82 " 6

cot ( "adjopp

"2

1.5"

43

sec ( "hypadj

"2.52

"54

csc ( "hypopp

"2.51.5

"53

tan ( "oppadj

"1.52

"34

cos ( "adjhyp

"2

2.5"

45

sin ( "opphyp

"1.52.5

"35

"2

2.5opp " &2.52 ! 22 " 1.5

The function values are the same since the triangles are similar and the corresponding sides are proportional.

©H

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7.

cot ( "adjopp

" 2&2

sec ( "hypadj

"3

2&2"

3&24

csc ( "hypopp

" 3

tan ( "oppadj

"1

2&2"

&24

cos ( "adjhyp

"2&2

3

" 31

sin ( "opphyp

"13

adj " &32 ! 12 " &8 " 2&2

8.

sin

cos

tan

csc

sec

cot ( "adjopp

"21

" 2

( "hypadj

"&52

( "hypopp

"&51

" &5

( "oppadj

"12

( "adjhyp

"2&5

"2&5

5

( "opphyp

"1&5

"&55

1

2

"hyp " &12 # 22 " &5

sin

cos

tan

csc

sec

cot ( "63

" 2

( "3&5

6"

&52

( "3&5

3" &5

( "36

"12

( "6

3&5"

2&5

"2&5

5

( "3

3&5"

1&5

"&55

3

6"

hyp " &32 # 62 " 3&5

The function values are the same because the triangles are similar, and corresponding sides are proportional.

cot ( "adjopp

"4&2

2" 2&2

sec ( "hypadj

"6

4&2"

3

2&2"

3&24

csc ( "hypopp

"62

" 3

tan ( "oppadj

"2

4&2"

1

2&2"

&24

cos ( "adjhyp

"4&2

6"

2&23

"

62

sin ( "opphyp

"26

"13

adj " &62 ! 22 " &32 " 4&2

The function values are the same since the triangles are similar and the corresponding sides are proportional.

©H

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10.

sin

cos

tan

csc

sec ( "hypadj

"&26

5

( "hypopp

"&26

1" &26

( "oppadj

"15

( "adjhyp

"5

&26"

5&2626

( "opphyp

"1

&26"

&2626

1

5

26"

hyp " &52 # 12 " &26

12.

sin

tan

csc

sec

cot ( "3

2&10"

3&1020

( "73

( "7

2&10"

7&1020

( "2&10

3

( "2&10

7

"

102

3

7

opp " &72 ! 32 " &40 " 2&10

9. Given:

csc ( "hypopp

"65

sec ( "hypadj

"6

&11"

6&1111

cot ( "adjopp

"&11

5

tan ( "oppadj

"5

&11"

5&1111

cos ( "adjhyp

"&11

6

adj " &11

52 # "adj#2 " 62

56

11

"

sin ( "56

"opphyp

11. Given:

csc ( "4

&15"

4&1515

cot ( "1

&15"

&1515

tan ( " &15

cos ( "14

sin ( "&15

4

opp " &15

"opp#2 # 12 " 42

15

"

4

1

sec ( " 4 "41

"hypadj

14.

sin

cos

tan

sec

cot ( "1

tan ("

&2734

( "1

cos ("

17&273

"17&273

273

( "oppadj

"4

&273"

4&273273

( "adjhyp

"&273

17

( "opphyp

"417

417

273"

adj " &172 ! 42 " &27313. Given:

csc ( "&10

3

cot ( "13

sec ( " &10

cos ( "&1010

sin ( "3&10

10

hyp " &10

32 # 12 " "hyp#2

3

1

10

"

tan ( " 3 "31

"oppadj

©H

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294 Chapter 4 Trigonometric Functions294 Chapter 4 Trigonometric Functions

16.

cos

tan

csc

sec

cot ( "1

tan ("

&553

( "1

cos ("

8&55

"8&55

55

( "1

sin ("

83

( "oppadj

"3

&55"

3&5555

( "adjhyp

"&55

8

adj " &82 ! 32 " &553

8

55

"

sin ( "38

15. Given:

csc ( "&97

4

sec ( "&97

9

tan ( "49

cos ( "9

&97"

9&9797

sin ( "4

&97"

4&9797

hyp " &97

42 # 92 " "hyp#2

97

"9

4

cot ( "94

"adjopp

Function (deg) (rad) Function Value((

26. tan1&3

*6

30%

24. sin&22

*4

45%

22. csc &2*4

45%

20. sec &2*4

45%

18. cos&22

*4

45%

Function (deg) (rad) Function Value

17. sin

19. tan

21. cot

23. cos

25. cot 1*4

45%

&32

*6

30%

&33

*3

60%

&3*3

60%

12

*6

30%

((

27. sin ( "1

csc (28. cos ( "

1sec (

29. tan ( "1

cot (30. csc ( "

1sin (

31. sec ( "1

cos (32. cot ( "

1tan ( 33. tan ( "

sin (cos (

34. cot ( "cos (sin (

35. sin2 ( # cos2 ( " 1 36. 1 # tan2 ( " sec2 ( 37. sin"90% ! (# " cos ( 38. cos"90% ! (# " sin (

39. tan"90% ! (# " cot ( 40. cot"90% ! (# " tan ( 41. sec"90% ! (# " csc ( 42. csc"90% ! (# " sec (

43.

(a)

(c) cos 30% " sin 60% "&32

tan 60% "sin 60%

cos 60%" &3

sin 60% "&32

, cos 60% "12

(b)

(d) cot 60% "cos 60%

sin 60%"

1&3

"&33

sin 30% " cos 60% "12

©H

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44.

(a)

(b)

(c)

(d) cot 30% "1

tan 30%"

3&3

"3&3

3" &3

cos 30% "sin 30%

tan 30%"

"1'2#"&3'3# "

3

2&3"

&3

2

cot 60% " tan"90% ! 60%# " tan 30% "&33

csc 30% "1

sin 30%" 2

sin 30% "12

, tan 30% "&33

46.

(a)

(b) cot ( "1

tan ("

12&6

"&612

cos ( "1

sec ("

15

sec ( " 5, tan ( " 2&6

45.

(a)

(b)

(c)

(d) sec"90º ! (# " csc ( " 3

tan ( "sin (cos (

"1'3

"2&2 #'3"

&24

cos ( "1

sec ("

2&23

sin ( "1

csc ("

13

csc ( " 3, sec ( "3&2

4

47.

(a)

(b)

sin & "&15

4

sin2 & "1516

sin2 & # $14%

2

" 1

sin2 & # cos2 & " 1

sec & "1

cos &" 4

cos & "14

(c)

(d) sin"90% ! &# " cos & "14

"&1515

"1

&15

"1'4

&15'4

cot & "cos &sin &

(c)

(d) sin ( " tan ( cos ( " "2&6 #$15% "

2&65

cot"90% ! (# " tan ( " 2&6

48.

(a)

(b)

(c)

(d) csc ' " &1 # cot2 ' "&1 #125

"&26

5

tan"90% ! '# " cot ' "15

sec2 ' " 1 # tan2 ' ! cos ' " 1

&1 # tan2 '"

1&1 # 25

" 1

&26"

&2626

cot ' "1

tan '"

15

"' lies in Quadrant I or III.#tan ' " 5

50. csc ( tan ( "1

sin ( )sin (cos (

"1

cos (" sec (49. tan ( cot ( " tan ($ 1

tan (% " 1

51. tan ( cos ( " $ sin (cos (% cos ( " sin ( 52. cot ( sin ( "

cos (sin (

sin ( " cos (

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.


53.

" sin2 (

" "sin2 ( # cos2 (# ! cos2 (

"1 # cos (#"1 ! cos (# " 1 ! cos2 (

55.

" csc ( sec ( "1

sin ()

1cos (

"1

sin ( cos (

sin (cos (

#cos (sin (

"sin2 ( # cos2 (

sin ( cos (

54.

" 1

"csc ( # cot (#"csc ( ! cot (# " csc2 ( ! cot2 (

56.

" 1 # cot2 ( " csc2 (

" 1 #cot (

"1'cot (#

tan ( # cot (tan (

"tan (tan (

#cot (tan (

57. (a)

(b) cos 87% ! 0.0523

sin 41% ! 0.6561

59. (a)

(b) csc 48º 7+ "1

sin"48 # 760#º ! 1.3432

sec 42% 12+ " sec 42.2% "1

cos 42.2%! 1.3499

58. (a)

(b) cot 71.5% "1

tan 71.5%! 0.3346

tan 18.5% ! 0.3346

60. (a)

(b) sec"8% 50+ 25, # "1

cos"8% 50+ 25, # ! 1.0120

! cos"8.840278# ! 0.9881

cos"8% 50+ 25, # " cos$8 #5060

#25

3600%

61. Make sure that your calculator is in radian mode.

(a)

(b) tan *

8! 0.4142

cot *

16"

1tan"*'16# ! 5.0273

63. (a)

(b) csc ( " 2 ! ( " 30% "*

6

sin ( "12

! ( " 30% "*

6

62. (a)

(b) cos"1.25# ! 0.3153

sec"1.54# "1

cos"1.54# ! 32.4765

64. (a)

(b) tan ( " 1 ! ( " 45% "*

4

cos ( "&22

! ( " 45% "*

4

65. (a)

(b) cot ( " 1 ! ( " 45% "*

4

sec ( " 2 ! ( " 60% "*

3

67. (a)

(b) sin ( "&22

! ( " 45% "*

4

csc ( "2&3

3 ! ( " 60% "

*

3

66. (a)

(b) cos ( "12

! ( " 60% "*

3

tan ( " &3 ! ( " 60% "*

3

68. (a)

(b)

cos ( "1&2

"&22

! ( " 45% "*

4

sec ( " &2

tan ( "3&3

" &3 ! ( " 60% "*

3

cot ( "&33

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69.

cos 30% "105

r ! r "

105cos 30%

"105&3'2

"210&3

" 70&3

tan 30% "y

105 ! y " 105 tan 30% " 105 )

&33

" 35&3

70.

sin 30% "y

15 ! y " 15 sin 30% " 15$1

2% "152

cos 30% "x

15 ! x " 15 cos 30% " 15 )

&32

"15&3

2

71.

sin 60% "y

16 ! y " 16 sin 60% " 16$&3

2 % " 8&3

cos 60% "x

16 ! x " 16 cos 60% " 16$1

2% " 8

72.

sin 60% "38r

! r "38

sin 60%"

38&3'2

"76&3

3

cot 60% "x

38 ! x " 38 cot 60% " 38 )

1&3

"38&3

3

73.

r2 " 202 # 202 ! r " 20&2

tan 45% "20x

! 1 "20x

! x " 20 74.

r2 " 102 # 102 ! r " 10&2

tan 45% "y

10 ! 1 "

y10

! y " 10

75.

r2 " "2&5#2 # "2&5#2 " 20 # 20 " 40 ! r " 2&10

tan 45% "2&5

x ! 1 "

2&5x

! x " 2&5

76.

r2 " "4&6#2 # "4&6#2 " 96 # 96 " 192 ! r " 8&3

tan 45% "y

4&6 ! 1 "

y4&6

! y " 4&6

77. (a)

(b) and Thus,

(c) feeth "6"21#

5" 25.2

65

"h21

.tan ( "h21

tan ( "65

6

h

516

"

78. (a)

(b)

(c) x " 30 sin 75% ! 28.98 meters

sin 75% "x

30

x

75°

30

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.


80.

Subtracting,

! 1.295 ! 1.3 miles. ! 1316.3499 ! 6.3138

h "13

cot 3.5% ! cot 9%

13h

" cot 3.5% ! cot 9%

cot 3.5% "13 # c

h

not drawn to scale

9°3.5°13 c

h

cot 9% "ch

79.

feetw " 100 tan 58% ! 160

tan 58% "w

100

tan ( "oppadj

81. (a)

(b)

(c) rate down the zip line

vertical rate506

"253

ft'sec

50&26

"25&2

3 ft'sec

L2 " 502 # 502 " 2 ) 502 ! L " 50&2 feet

tan ( "5050

" 1 ! ( " 45% 82. (a)

(b) feet above sea level

(c) minutes to reach top

Vertical rate

feet per minute ! 173.8

"519.3

"896.5'300#

896.5300

1693.5 ! 519.3 " 1174.2

x " 896.5 sin 35.4% ! 519.3 feet

sin 35.4% "x

896.5

83.

"x1, y1# " "28&3, 28#

x1 " cos 30%"56# "&32

"56# " 28&3

cos 30% "x1

56

y1 " "sin 30%#"56# " $12%"56# " 28

sin 30% "y1

56

30°

56

( , )x y1 1

"x2, y2# " "28, 28&3 #

x2 " "cos 60%#"56# " $12%"56# " 28

cos 60% "x2

56

y2 " sin 60%"56# " $&32 %"56# " 28&3

sin 60º "y2

56

60°

56

( , )x y2 2

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.


84.

csc 20% "10y

! 2.92sec 20% "10x

! 1.06

cot 20% "xy

! 2.75tan 20% "yx

! 0.36

cos 20% "x

10! 0.94sin 20% "

y10

! 0.34

x ! 9.397, y ! 3.420

86. False

" &2 $ 1 sin 45% # cos 45% "&22

#&22

87. True

for all (1 # cot2 ( " csc2 (

85. True

sin 60% csc 60% " sin 60% 1

sin 60%" 1

88. No. so you can find ± sec (.tan2 ( # 1 " sec2 (,

(b) Sine and tangent are increasing, cosine isdecreasing.

(c) In each case, tan ( "sin (cos (

.

89. (a)

0 0.3420 0.6428 0.8660 0.9848

1 0.9397 0.7660 0.5000 0.1736

0 0.3640 0.8391 1.7321 5.6713tan (

cos (

sin (

80%60%40%20%0%(

90.

1 0.9397 0.7660 0.5000 0.1736

1 0.9397 0.7660 0.5000 0.1736sin"90% ! (#

cos (

80%60%40%20%0%(are complementary angles.( and 90% ! (

cos ( " sin"90% ! (#

91.

!6 6

!1

7 93.

!6 6

!1

7

95.

!2 10

!4

4 97.

!2 10

!4

4

92.!3 3

!3

1 94.!6 6

!6

2

96.

!2 10

!4

4 98.

!2 10

!4

4

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.

Section 4.4 Trigonometric Functions of Any Angle


! Know the Definitions of Trigonometric Functions of Any Angle.

If is in standard position, a point on the terminal side and then:

! You should know the signs of the trigonometric functions in each quadrant.

! You should know the trigonometric function values of the quadrant angles

! You should be able to find reference angles.

! You should be able to evaluate trigonometric functions of any angle. (Use reference angles.)

! You should know that the period of sine and cosine is

! You should know which trigonometric functions are odd and even.

Even: cos x and sec x

Odd: sin x, tan x, cot x, csc x

2!.

0, !

2, !, and

3!

2.

cot " #xy, y $ 0tan " #

yx, x $ 0

sec " #rx, x $ 0cos " #

xr

csc " #ry, y $ 0sin " #

yr

r # !x2 % y2 $ 0,"x, y#"

Vocabulary Check

1. 2. 3. 4.

5. 6. 7. referencecot "cos "

rx

yx

csc "yr

1. (a)

cot " #xy

#43

tan " #yx

#34

sec " #rx

#54

cos " #xr

#45

csc " #ry

#53

sin " #yr

#35

r # !16 % 9 # 5

"x, y# # "4, 3# (b)

cot " #xy

#815

tan " #yx

#158

sec " #rx

# &178

cos " #xr

# &817

csc " #ry

# &1715

sin " #yr

# &1517

r # !64 % 225 # 17

"x, y# # "&8, &15#

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Section 4.4 Trigonometric Functions of Any Angle 301

2. (a)

cot " #xy

#12&5

# &125

sec " #rx

#1312

csc " #ry

#13&5

# &135

tan " #yx

#&512

# &512

cos " #xr

#1213

sin " #yr

#&513

# &513

r # !122 % "&5#2 # 13

x # 12, y # &5 (b)

cot " #xy

#&11

# &1

sec " #rx

#!2&1

# &!2

csc " #ry

#!21

# !2

tan " #yx

#1

&1# &1

cos " #xr

#&1!2

# &!22

sin " #yr

#1!2

#!22

r # !"&1#2 % 12 # !2

x # &1, y # 1

4. (a)

cot " #xy

#31

# 3

sec " #rx

#!10

3

csc " #ry

#!10

1# !10

tan " #yx

#13

cos " #xr

#3

!10#

3!1010

sin " #yr

#1

!10#

!1010

r # !32 % 12 # !10

x # 3, y # 1 (b)

cot " #xy

#2

&4# &

12

sec " #rx

#2!5

2# !5

csc " #ry

#2!5&4

# &!52

tan " #yx

#&42

# &2

cos " #xr

#2

2!5#

!55

sin " #yr

#&4

2!5# &

2!55

r # !22 % "&4#2 # 2!5

x # 2, y # &4

3. (a)

cot " #xy

# !3tan " #yx

#!33

sec " #rx

#&2!3

3cos " #

xr

#&!3

2

csc " #ry

# &2sin " #yr

# &12

r # !3 % 1 # 2

"x, y# # "&!3, &1# (b)

cot " #xy

# &1tan " #yx

# &1

sec " #rx

# &!2cos " #xr

# &!22

csc " #ry

# !2sin " #yr

#!22

r # !4 % 4 # 2!2

"x, y# # "&2, 2#

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.


5.

cot " #xy

#724

tan " #yx

#247

sec " #rx

#257

cos " #xr

#725

csc " #ry

#2524

sin " #yr

#2425

r # !49 % 576 # 25

"x, y# # "7, 24#

7.

cot " #xy

# &512

sec " #rx

#135

csc " #ry

# &1312

tan " #yx

# &125

cos " #xr

#513

sin " #yr

# &1213

r # !52 % "&12#2 # !25 % 144 # !169 # 13

"x, y# # "5, &12#

6.

sin cos

tan csc

sec cot " #xy

#815

" #rx

#178

" #ry

#1715

" #yx

#158

" #xr

#817

" #yr

#1517

r # !82 % 152 # 17

x # 8, y # 15,

8.

sin

cos

tan

csc

sec

cot " #xy

# &125

" #rx

# &1312

" #ry

#135

" #yx

#10

&24# &

512

" #xr

#&2426

#&1213

" #yr

#1026

#513

r # !"&24#2 % "10#2 # 26

x # &24, y # 10

9.

cot " #xy

# &25

sec " #rx

# &!29

2

csc " #ry

#!29

5

tan " #yx

# &52

cos " #xr

# &2!29

29

sin " #yr

#5!29

29

r # !16 % 100 # 2!29

"x, y# # "&4, 10# 10.

sin

cos

tan

csc

sec

cot " #xy

#56

" #rx

# &!61

5

" #ry

# &!61

6

" #yx

#&6&5

#65

" #xr

#&5!61

#&5!61

61

" #yr

#&6!61

#&6!61

61

r # !"&5#2& % "&6#2 # !61

x # &5, y # &6

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.


11.

cot " #xy

# &54

sec " #rx

# &&!41

5

csc " #ry

#!41

4

tan " #yx

#8

&10# &

45

cos " #xr

#&10

2!41# &

5!4141

sin " #yr

#8

2!41#

4!4141

r # !"&10#2 % 82 # !164 # 2!41

"x, y# # "&10, 8# 12.

sin

cos

tan

csc

sec

cot " #xy

# &13

" #rx

# !10

" #ry

# &!10

3

" #yx

#&93

# &3

" #xr

#3

3!10#

!1010

" #yr

#&9

3!10#

&3!1010

r # !32 % "&9#2 # !90 # 3!10

x # 3, y # &9

13. lies in Quadrant III or Quadrant IV.

lies in Quadrant II or Quadrant III.

and lies in Quadrant III.cos " < 0 ! "sin " < 0

cos " < 0 ! "

sin " < 0 ! "

15. lies in Quadrant I or Quadrant III.

lies in Quadrant I or Quadrant IV.

and lies in Quadrant I.cos " > 0 ! "cot " > 0

cos " > 0 ! "

cot " > 0 ! "

17.

in Quadrant II

cot " #xy

# &43

tan " #yx

# &34

sec " #rx

# &54

cos " #xr

# &45

csc " #ry

#53

sin " #yr

#35

! x # &4"

sin " #yr

#35

! x2 # 25 & 9 # 16

14. and

and

Quadrant IV

xy

< 0rx

> 0

cot " < 0sec " > 0

16. and

and

Quadrant III

ry

< 0yx

> 0

csc " < 0tan " > 0

18.

in Quadrant III

sin csc

cos sec

tan cot " #43

" #yx

#34

" # &54

" #xr

# &45

" # &53

" #yr

# &35

! y # &3"

cos " #xr

#&45

! $y$ # 3

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.


19.

cot " #xy

# &815

sec " #rx

#178

cos " #xr

#817

csc " #ry

# &1715

sin " #yr

# &1517

tan " #yx

#&15

8 ! r # 17

sin " < 0 ! y < 0 20. csc

cot

sin csc

cos sec

tan cot " # &!15" #yx

# &!1515

" # &4!15

15" #

xr

# &!15

4

" # 4" #yr

#14

" < 0 ! x # &!15

" #ry

#41

! x # ±!15

21.

cot " #xy

# &!33

tan " #yx

# &!3

sec " #rx

# &2cos " #xr

# &12

csc " #ry

#2!3

3sin " #

yr

#!32

sin " " 0 ! y # !3

sec " #rx

#2

&1 ! y2 # 4 & 1 # 3

23. is undefined

is undefined.

is undefined.cot "tan " #yx

#0x

# 0

sec " #rx

# &1cos " #xr

#&rr

# &1

csc " #ry

sin " #yr

# 0

!2

# " #3!2

! " # !, y # 0, x # &r

! " # n!.cot "

22. and

is undefined.

is undefined.cot "

sec " # &1

csc "

tan " #sin "cos "

# 0

cos " # cos ! # &1

!2

# " #3!2

! " # !sin " # 0

24. is undefined and

is undefined.

is undefined.

cot " # 0

sec "

csc " # &1

tan "

cos " # 0

sin " # &1

! # " # 2! ! " #3!2

.tan "

25. To find a point on the terminal side of use any point on the line that lies inQuadrant II. is one such point.

cot " # &1tan " # &1

sec " # &!2cos " # &1!2

# &!22

csc " # !2sin " #1!2

#!22

x # &1, y # 1, r # !2

"&1, 1#y # &x",

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.


27. To find a point on the terminal side of use anypoint on the line that lies in Quadrant III.

is one such point.

cot " #&1&2

#12

sec " #!5&1

# &!5

csc " #!5&2

# &!52

tan " #&2&1

# 2

cos " # &1!5

# &!55

sin " # &2!5

# &2!5

5

x # &1, y # &2, r # !5

"&1, &2#y # 2x

",26. Quadrant III,

sin

cos

tan

csc

sec

cot " #xy

#&x

"&1%3# x# 3

" #rx

#"!10x#%3

&x# &

!103

" #ry

#"!10x#%3"&1%3#x

# &!10

" #yx

#"&1%3#x

&x#

13

" #xr

#&x

"!10x#%3# &

3!1010

" #yr

#"&x%3#

"!10x#%3# &

!1010

r #!x2 %19

x2 #!10x

3

x > 0&&x, &13

x'

28.

Quadrant IV,

sin csc

cos sec

tan cot " # &34

" #yx

#"&4%3# x

x# &

43

" #53

" #xr

#x

"5%3#x#

35

" # &54

" #yr

#"&4%3#x"5%3# x

# &45

r #!x2 %169

x2 #53

x

x > 0&x, &43

x'

4x % 3y # 0 ! y # &43

x

29.

sec ! #rx

#1

&1# &1

"x, y# # "&1, 0# 31.

#0

&1# 0 cot&3!

2 ' #xy

"x, y# # "0, &1#

33.

sec 0 #rx

#11

# 1

"x, y# # "1, 0#

35.

undefined cot ! #xy

# &10

!

"x, y# # "&1, 0#

30. undefined

since corresponds to "0, 1#.!

2

tan !

2#

yx

#10

!

32. undefinedcsc 0 #ry

#10

! 34. csc 3!2

#1

sin 3!2

#1

&1# &1

36. csc !2

#1

sin !2

#11

# 1

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.


37.

"' # 180( & 120( # 60(

x

y

120!=!60!=! "

" # 120(

39. is coterminal with

x

y

#235!=!

55!=! "

"' # 225( & 180( # 45(

225(." # &135(

38.

x

y

225!=!

45!=! "

"' # 225( & 180( # 45(

40.

x

y

#330!=! 30!=! "

"' # &330( % 360( # 30(

41.

x

y

=

! =

$3

$3

5

! "

" #5!3

, "' # 2! &5!3

#!3


x

y

=! =

$6 $

65#

! "

"' #7!6

& ! #!6

7!6

." # &5!6

42.

x

y

= $4

! = $4

3! "

"' # ! &3!4

#!4

44.

x

y

= $3 ! = $

32#

! "

"' #&2!

3% ! #

!3

45.

"' # 208( & 180( # 28(= 208°

x

y

!!

" = 28°

" # 208( 46.

"' # 360( & 322( # 38(

x

y

= 322°

!

!

" = 38°

" # 322(

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.


47.

"' # 360( & 292( # 68(

x

y

= #292° !! " = 68°

" # &292( 48. lies inQuadrant III.

Reference angle:180( & 165( # 15(

x

y

#165!=

! " 15!=

!

" # &165(


x

y

$5

11

$5

=

!

!

" =

"' #!5

!5

." #11!

5

51. lies in Quadrant III.

Reference angle:

x

y

#1.8=! " 1.342=

!

! & 1.8 ( 1.342

" # &1.8

50.

x

y

$7

17

$7

3!

!

" =

=

"' #17!

7&

14!7

#3!7

52. lies in Quadrant IV.

Reference angle:

x

y

! "= 1.358

! = 4.5

4.5 & ! ( 1.358

"

53. Quadrant III

tan 225( # tan 45( # 1

cos 225( # &cos 45( # &!22

sin 225( # &sin 45( # &!22

"' # 45(,

55. is coterminal with Quadrant IV.

tan"&750(# # &tan 30( # &!33

cos"&750(# # cos 30( #!32

sin"&750(# # &sin 30( # &12

"' # 360( & 330( # 30(

330(," # &750(

54. Quadrant IV

sin

cos

tan 300( # &tan 60( # &!3

300( # cos 60( #12

300( # &sin 60( # &!32

" # 300(, "' # 360( & 300( # 60(,

56. Quadrant III

tan"&495(# # tan 45( # 1

cos"&495(# # &cos 45( # &!22

sin"&495(# # &sin 45( # &!22

" # &495(, "' # 45(,

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.


57. Quadrant IV

tan&5!3 ' # &tan&!

3' # &!3

cos&5!3 ' # cos&!

3' #12

sin&5!3 ' # &sin&!

3' # &!32

"' # 2! &5!3

#!3

" #5!3

, 58.

sin

cos

tan 3!4

# &1

3!

4# &

!22

3!

4#

!22

" #3!

4, "' # ! &

3!

4#

!

4

59. Quadrant IV

tan&&!

6' # &tan !

6# &

!33

cos&&!

6' # cos !

6#

!32

sin&&!

6' # &sin !

6# &

12

"' #!

6,

61. Quadrant II

tan 11!

4# &tan

!

4# &1

cos 11!

4# &cos

!

4# &

!22

sin 11!

4# sin

!

4#

!22

"' #!

4,

60. ,

sin

cos

tan&&4!

3 ' # &!3

&&4!

3 ' #&12

&&4!

3 ' #!32

"' #!

3" #

&4!

3


in Quadrant III.

sin

cos

tan 10!

3# tan

!

3# !3

10!

3# &cos

!

3# &

12

10!

3# &sin

!

3# &

!32

"' #4!

3& ! #

!

3

4!

3." #

10!

3


Quadrant III

tan&&17!

6 ' # tan&!6' #

!33

cos&&17!

6 ' # &cos&!6' # &

!32

sin&&17!

6 ' # &sin&!6' # &

12

"' #7!6

& ! #!6

,

7!6

." # &17!

664.

sin

cos

tan&&20!

3 ' # !3

&&20!

3 ' #&12

&&20!

3 ' #&!3

2

" #&20!

3, "' #

!3

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65.

in Quadrant IV.

cos " #45

cos " > 0

cos2 " #1625

cos2 " # 1 &925

cos2 " # 1 & &&35'

2

cos2 " # 1 & sin2 "

sin2 " % cos2 " # 1

sin " # &35

66.

sin " #1

csc "#

1!10

#!1010

csc " #1

sin "

!10 # csc "

csc " > 0 in Quadrant II.

10 # csc2 "

1 % "&3#2 # csc2 "

1 % cot2 " # csc2 "

cot " # &3

67.

cot " # &!3

cot " < 0 in Quadrant IV.

cot2 " # 3

cot2 " # "&2#2 & 1

cot2 " # csc2 " & 1

1 % cot2 " # csc2 "

csc " # &2

69.

tan " #!65

4

tan " > 0 in Quadrant III.

tan2 " #6516

tan2 " # &&94'

2& 1

tan2 " # sec2 " & 1

1 % tan2 " # sec2 "

sec " # &94

68.

sec " #1

5%8#

85

cos " #1

sec " ! sec " #

1cos "

cos " #58

70.

Quadrant IV ! csc " # &!41

5

csc " # ±!41

5

1 %1625

# csc2 "

1 % cot2 " # csc2 "

tan " # &54

! cot " # &45

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71. is in Quadrant II.

cot " #1

tan "#

&!212

sec " #1

cos "#

&5!21

#&5!21

21

csc " #1

sin "#

52

tan " #sin "c)s "

#2%5

&!21%5#

&2!21

#&2!21

21

cos " # &!1 & sin2 " # &!1 &425

# &!21

5

sin " #25

and cos " < 0 ! "

72. is in Quadrant III.

sec " #1

cos "# &

73

csc " #1

sin "#

&72!10

#&7!10

20

cot " #1

tan "#

32!10

#3!10

20

tan " #sin "c)s "

#&2!10%7

&3%7#

2!103

sin " # &!1 & cos2 " # &!1 &949

#&!40

7#

&2!107

cos " # &37

and sin " < 0 ! "

73. is in Quadrant II.

cot " #1

tan "# &

14

csc " #1

sin "#

174!17

#!17

4

sin " # tan " cos " # "&4#&&!1717 ' #

4!1717

cos " #1

sec "#

&1!17

#&!17

17

sec " # &!1 % tan2 " # &!1 % 16 # &!17

tan " # &4 and cos " < 0 ! " 74. is in Quadrant II.

csc " #1

sin "#

26!26

# !26

sin " # tan " cos " # &&15'&&5!26

26 ' #!2626

cos " #1

sec "#

&5!26

#&5!26

26

sec " # &!1 % tan2 " # &!1 %125

#&!26

5

tan " #1

cot "#

&15

cot " # &5 and sin " > 0 ! "

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78. sec 235( #1

cos 235(( &1.7434

75. is in Quadrant IV.

cot " #1

tan "#

&!52

tan " #sin "cos "

#&2%3!5%3

#&2!5

#&2!5

5

sec " #1

cos "#

3!5

#3!5

5

cos " # !1 & sin2 " #!1 &49

#!53

sin " #1

csc "# &

23

csc " # &32

and tan " < 0 ! " 76. is in Quadrant III.

cot " #1

tan "#

3!7

#3!7

7

tan " #sin "cos "

#&!7%4&3%4

#!73

csc " #1

sin "# &

4!7

#&4!7

7

sin " # &!1 & cos2 " # &!1 &916

# &!74

cos " #1

sec "# &

34

sec " # &43

and cot " > 0 ! "

77. sin 10( ( 0.1736 79. tan 245( ( 2.1445

81. cos"&110(# ( &0.342080. csc 320( #1

sin 320(# &1.5557 82.

( &1.1918

cot "&220(# #1

tan"&220(#

84. csc 0.33 #1

sin 0.33( 3.0860

86. tan 11!

9( 0.8391 88. cos&&15!

14 ' ( &0.9749

83.

( 5.7588

sec"&280(# #1

cos"&280(# 85. tan&2!9 ' ( 0.8391

87.

( &2.9238

csc&&8!9 ' #

1

sin&&8!9 '

89. (a) reference angle is or

and is in Quadrant I or Quadrant II.

Values in degrees:

Values in radian:

(b) reference angle is or

and is in Quadrant III or Quadrant IV.

Values in degrees:

Values in radians:7!

6,

11!

6

210(, 330(

"!

6

30(sin " # &12

!

!

6,

5!

6

30(, 150(

"!

6

30(sin " #12

! 90. (a) cos reference angle is 45º or

and is in Quadrant I or IV.

Values in degrees:

Values in radians:

(b) cos reference angle is or

and is in Quadrant II or III.

Values in degrees:


4,

5!

4

135(, 225(

"!

4

45(" # &!22

!

!

4,

7!

4

45(, 315(

"!

4

" #!22

!

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92. (a) csc

Reference angle is

Values in degrees:

Values in radians:

(b) csc

Reference angle is

Values in degrees:

Values in radians:!6

, 5!6

30(, 150(

!6

or 30(.

sin " #12

" # 2 !

5!4

, 7!4

225(, 315(

45( or !4

.

sin " #&1!2

" # &!2 ! 91. (a) reference angle is or

and is in Quadrant I or Quadrant II.

Values in degrees:

Values in radians:


and is in Quadrant II or Quadrant IV.

Values in degrees:


4,

7!

4

135(, 315(

"!

4

45(cot " # &1 !

!

3,

2!

3

60(, 120(

"!

3

60(csc " #2!3

3 !

93. (a) reference angle is or

and is in Quadrant II or Quadrant III.

Values in degrees:

Values in radians:


and is in Quadrant II or Quadrant III.

Values in degrees:

Values in radians:2!3

, 4!3

120(, 240(

"

60(,!3

cos " # &12

!

5!6

, 7!6

150(, 210(

"30(,

!6

sec " # &2!3

3 ! 94. (a)

Reference angle is

Values in degrees:

Values in radians:

(b) Values in degrees:

Values in radians:!4

or 7!4

45( or 315(

5!6

, 11!

6

150(, 330(

!6

or 30(.

cot " # &!3 ! cos "sin "

# &!3

95. (a)

(b)

(c) )cos 30(*2 # &!32 '2

#34

cos 30( & sin 30( #!3 & 1

2

f""# % g""# # sin 30( % cos 30( #12

%!32

#1 % !3

2

96. (a)

(b)

(c) )cos 60(*2 # &12'

2#

14

cos 60( & sin 60( #12

&!32

#1 & !3

2

f""# % g""# # sin 60( % cos 60( #!32

%12

#!3 % 1

2

(d)

(e)

(f) cos"&30(# # cos 30( #!32

2 sin 30( # 2&12' # 1

sin 30( cos 30( # &12'&!3

2 ' #!34

(d)

(e)

(f) cos"&60(# # cos 60( #12

2 sin 60( # 2!32

# !3

sin 60( cos 60( # &!32

'&12' #

!34

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97. (a)

(b)

(c) )cos 315(*2 # &!22 '2

#12

cos 315( & sin 315( #!22

& &&!22 ' # !2

f""# % g""# # sin 315( % cos 315( # &!22

%!22

# 0

98. (a)

(b)

(c)

(d) sin 225( cos 225( # &&!22 '&&!2

2 ' #12

)cos 225(*2 # &&!22 '2

#12

cos 225( & sin 225( #&!2

2& &&!2

2 ' # 0

f""# % g""# # sin 225( % cos 225( # &!22

&!22

# &!2

(d)

(e)

(f) cos"&315(# # cos"315(# #!22

2 sin 315( # 2&&!22 ' # &!2

sin 315( cos 315( # &&!22 '&!2

2 ' # &12

(e)

(f) cos"&225(# # cos"225(# # &!22

2 sin 225( # 2&&!22 ' # &!2

99. (a)

(b)

(c)

(d) sin 150( cos 150( #12 *

&!32

#&!3

4

)cos 150(*2 # &&!32 '2

#34

cos 150( & sin 150( #&!3

2&

12

#&1 & !3

2

f""# % g""# # sin 150( % cos 150( #12

%&!3

2#

1 & !32

(e)

(f ) cos"&150(# # cos"150(# #&!3

2

2 sin 150( # 2&12' # 1

100. (a)

(b)

(c)

(d) sin 300( cos 300( # &&!32 '&1

2' #&!3

4

)cos 300(*2 # &12'

2#

14

cos 300( & sin 300( #12

& &&!32 ' #

1 % !32

f""# % g""# # sin 300( % cos 300( #&!3

2%

12

#1 & !3

2

(e)

(f) cos"&300(# # cos"300(# #12

2 sin 300( # 2&&!32 ' # &!3

101. (a)

(b)

(c)

(d) sin 7!6

cos 7!6

# &&12'&&

!32 ' #

!34

+cos 7!6 ,2

# &&!32 '2

#34

cos 7!6

& sin 7!6

#&!3

2& &&

12' #

1 & !32

f""# % g""# # sin 7!6

% cos 7!6

# &12

&!32

#&1 & !3

2

(e)

(f) cos&&7!6 ' # cos&7!

6 ' #&!3

2

2 sin 7!6

# 2&&12' # &1

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102. (a)

(b)

(c)

(d) sin 5!6

cos 5!6

# &12'&&!3

2 ' #&!3

4

+cos 5!6 ,2

# &&!32 '2

#34

cos 5!6

& sin 5!6

#&!3

2&

12

#&1 & !3

2

f""# % g""# # sin 5!6

% cos 5!6

#12

%&!3

2#

1 & !32

(e)

(f) cos&&5!6 ' # cos&5!

6 ' #&!3

2

2 sin 5!6

# 2&12' # 1

103. (a)

(b)

(c)

(d) sin 4!3

cos 4!3

# &&!32 '&&

12' #

!34

+cos 4!3 ,2

# &&12'

2#

14

cos 4!3

& sin 4!3

# &12

& &&!32 ' #

!3 & 12

f""# % g""# # sin 4!3

% cos 4!3

#&!3

2&

12

#&1 & !3

2

(e)

(f) cos&&4!3 ' # cos&4!

3 ' # &12

2 sin 4!3

# 2&&!32 ' # &!3

104. (a)

(b)

(c)

(d) sin 5!3

cos 5!3

# &&!32 '&1

2' #&!3

4

+cos 5!3 ,2

# &12'

2#

14

cos 5!3

& sin 5!3

#12

& &&!32 ' #

1 % !32

f""# % g""# # sin 5!3

% cos 5!3

#&!3

2%

12

#1 & !3

2

(e)

(f) cos&&5!3 ' # cos&5!

3 ' #12

2 sin 5!3

# 2&&!32 ' # &!3

105. (a)

(b)

(c) )cos 270(*2 # 02 # 0

cos 270( & sin 270( # 0 & "&1# # 1

f""# % g""# # sin 270( % cos 270( # &1 % 0 # &1 (d)

(e)

(f) cos"&270(# # cos"270(# # 0

2 sin 270( # 2"&1# # &2

sin 270( cos 270( # "&1#"0# # 0

106. (a)

(b)

(c) )cos 180(*2 # "&1#2 # 1

cos 180( & sin 180( # &1 & 0 # &1

f""# % g""# # sin 180( % cos 180( # 0 & 1 # &1 (d)

(e)

(f) cos"&180(# # cos"180(# # &1

2 sin 180( # 2"0# # 0

sin 180( cos 180( # 0"&1# # 0

107. (a)

(b)

(c) +cos 7!2 ,2

# 02 # 0

cos 7!2

& sin 7!2

# 0 & "&1# # 1

f""# % g""# # sin 7!2

% cos 7!2

# &1 % 0 # &1 (d)

(e)

(f) cos&&7!2 ' # cos&7!

2 ' # 0

2 sin 7!2

# 2"&1# # &2

sin 7!2

cos 7!2

# "&1#"0# # 0

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108. (a)

(b)

(c) +cos 5!2 ,2

# 02 # 0

cos 5!2

& sin 5!2

# 0 & 1 # &1

f""# % g""# # sin 5!2

% cos 5!2

# 1 % 0 # 1 (d)

(e)

(f) cos&&5!2 ' # cos&5!

2 ' # 0

2 sin 5!2

# 2"1# # 2

sin 5!2

cos 5!2

# "1#"0# # 0

109.

(a) January:

(b) July:

(c) December: t # 12 ! T ( 31.75(

t # 7 ! T # 70(

t # 1 ! T # 49.5 % 20.5 cos&! "1#6

&7!6 ' # 29(

T # 49.5 % 20.5 cos&! t6

&7!6 '

110.

(a) thousand

(b) thousand

(c) thousand

(d) thousand

Answers will vary.

S"6# ( 25.8

S"5# ( 27.5

S"14# ( 33.0

S"1# # 23.1 % 0.442"1# % 4.3 sin&!6' ( 25.7

S # 23.1 % 0.442t % 4.3 sin&!t6 ', t # 1 $ Jan. 2004

111. sin

(a)

miles

(b)

miles

(c)

milesd #6

sin 120(( 6.9

" # 120(

d #6

sin 90(#

61

# 6

" # 90(

d #6

sin 30(#

6"1%2# # 12

" # 30(

" #6d

! d #6

sin "

113. True. The reference angle for isand sine is positive

in Quadrants I and II."' # 180( & 151( # 29(,

" # 151(

112. As increases from to x decreases from 12 cm to 0 cm and y increases from 0 cm to 12 cm. Therefore, increases from 0to 1, and decreases from 1 to 0.Thus, begins at 0 and increases without bound. When the tangent is undefined.

" # 90(,tan " # y%x

cos " # x%12sin " # y%12

90(,0("

114. False. and

cot&&!4' # &1

&cot&3!4 ' # &"&1# # 1

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115. (a)

(b) It appears that sin " # sin"180( & "#.

0 0.3420 0.6428 0.8660 0.9848

0 0.3420 0.6428 0.8660 0.9848sin"180( & "#

sin "

80(60(40(20(0("

116.

Patterns and conclusions may vary.

Function

Domain All reals except

Range

Evenness No Yes No

Oddness Yes No Yes

Period

Zeros n!!2

% n!n!

!2!2!

"&+, +#)&1, 1*)&1, 1*

!2

% n!"&+, +#"&+, +#

tan xcos xsin x

Function

Domain All reals except All reals except All reals except

Range

Evenness No Yes No

Oddness Yes No Yes

Period

Zeros None None!2

% n!

!2!2!

"&+, +#"&+, &1* ! )1, +#"&+, &1* ! )1, +#

n!!2

% n!n!

cot xsec xcsc x

118.

x # &179 ( &1.889

9x # &17

44 & 9x # 61

120.

x ( 1.186, &1.686

x #&1 ±!1 % 4"4#"2#

4#

&1 ±!334

2x2 % x & 4 # 0

117.

x # 7

3x # 21

3x & 7 # 14

119.

x ( 3.449, x ( &1.449

x #2 ±!4 % 20

2# 1 ±!6

x2 & 2x & 5 # 0

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Section 4.5 Graphs of Sine and Cosine Functions

Section 4.5 Graphs of Sine and Cosine Functions 317

121.

x ! !5.908, 4.908

x "!1 ±"1 # 4#29$

2"

!1 ±"1172

x2 # x ! 29 " 0

27 " #x ! 1$#x # 2$

3

x ! 1"

x # 29

123.

x " 3 ! log4 726 " 3 !ln 726ln 4

! !1.752

3 ! x " log4 726

43!x " 726

125.

x " e!6 ! 0.002479 ! 0.002

ln x " !6

122.

( extraneous)x " 0x " 6

x#x ! 6$ " 0

x2 ! 6x " 0

10x " x2 # 4x

5x

"x # 4

2x

124.

x "12

ln 86 ! 2.227

2x " ln 86

86 " e2x

90 " 4 # e2x

4500

4 # e2x " 50

126. ! x # 10 " e2 ! x " e2 ! 10 ! !2.611ln "x # 10 " 12 ln#x # 10$ " 1 ! ln#x # 10$ " 2

! You should be able to graph and

! Amplitude:

! Period:

! Shift: Solve and

! Key increments: (period)14

bx ! c " 2$.bx ! c " 0

2$

%b%

%a%y " a cos#bx ! c$.y " a sin#bx ! c$

Vocabulary Check

1. amplitude 2. one cycle 3. 4. phase shift2$b

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1.

(a) -intercepts:

(b) -intercept:

(c) Increasing on:

Decreasing on:

(d) Relative maxima:

Relative minima: &!$2

, !1', &3$2

, !1'&!

3$2

, 1', &$2

, 1'&!

3$2

, !$2', &$

2,

3$2 '

&!2$, !3$2 ', &!

$2

, $2', &3$

2, 2$'

#0, 0$y

#!2$, 0$, #!$, 0$, #0, 0$, #$, 0$, #2$, 0$x

f#x$ " sin x

2.

(a) -intercepts:

(b) -intercept:

(c) Increasing on:

Decreasing on:

(d) Relative maxima:

Relative minima: #!$, !1$, #$, !1$

#!2$, 1$, #0, 1$, #2$, 1$

#!2$, !$$, #0, $$

#!$, 0$, #$, 2$$

#0, 1$y

&!3$2

, 0', &!$2

, 0', &$2

, 0', &3$2

, 0'x

f#x$ " cos x

7.

Period:

Amplitude: %23% "23

2$

$" 2

y "23

sin $x

3.

Period:

Amplitude: %3% " 3

2$

2" $

y " 3 sin 2x

5.

Period:

Amplitude: %52% "52

2$

1(2" 4$

y "52

cos x2

Xmin = -2Xmax = 2Xscl = Ymin = -4Ymax = 4Yscl = 1

$(2$$

Xmin = -4Xmax = 4Xscl = Ymin = -3Ymax = 3Yscl = 1

$$$

4.

Period:

Amplitude: %a% " 2

2$

b"

2$

3

y " 2 cos 3x

6.

Period:

Amplitude: %a% " %!3% " 3

2$

b"

2$

#1(3$ " 6$

y " !3 sin x3

8.

Period:

Amplitude: %a% "32

2$

b"

2$

#$(2$ " 4

y "32

cos $x2

Xmin = -Xmax = Xscl = Ymin = -3Ymax = 3Yscl = 1

$(4$$

Xmin = -Xmax = Xscl = Ymin = -4Ymax = 4Yscl = 1

$6$6$

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Section 4.5 Graphs of Sine and Cosine Functions 319

9.

Period:

Amplitude: %!2% " 2

2$

1" 2$

y " !2 sin x 10.

Period:

Amplitude: %a% " %!1% " 1

2$b

"2$2(5

" 5$

y " !cos 2x5

11.

Period:

Amplitude: %14% "14

2$

2(3" 3$

y "14

cos 2x3

13.

Period:

Amplitude: %13% "13

2$4$

"12

y "13

sin 4$x

15.

The graph of is a horizontal shift to the right units of the graph of (a phase shift).f$

g

g#x$ " sin#x ! $$

f #x$ " sin x

17.

The graph of is a reflection in the axis of thegraph of f.

x-g

g#x$ " !cos 2x

f #x$ " cos 2x

19.

The graph of has five times the amplitude of and reflected in the axis.x-

f,g

g#x$ " !5 cos x

f#x$ " cos x

21.

The graph of is a vertical shift upward of fiveunits of the graph of f.

g

g#x$ " 5 # sin 2x

f #x$ " sin 2x

12.

Period:

Amplitude: %a% "52

2$

b"

2$

1(4" 8$

y "52

cos x4

14.

Amplitude: %a% "34

Period: 2$b

"2$

$(12" 24

y "34

cos $ x12

16.

g is a horizontal shift of f units to the left.$

f#x$ " cos x, g#x$ " cos#x # $$

18.

g is a reflection of f about the y-axis.(or, about the -axis)x

f #x$ " sin 3x, g#x$ " sin#!3x$

20.

The amplitude of g is one-half that of f. g is areflection of f in the x-axis.

f#x$ " sin x, g#x$ " !12 sin x

22.

g is a vertical shift of f six units downward.

f #x$ " cos 4x, g#x$ " !6 # cos 4x

24. The period of g is one-half the period of f.23. The graph of has twice the amplitude as the graphof The period is the same.f.

g

25. The graph of is a horizontal shift units to theright of the graph of f.

$g 26. Shift the graph of f two units upward to obtain thegraph of g.

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27.

Period:

Amplitude: 1

Period:

Amplitude:

!2

!3

!4

1

4

x

y

"2

"23!

f

g

%!4% " 4

2$

g#x$ " !4 sin x

2$

f #x$ " sin x

29.

Period:

Amplitude: 1

is a vertical shift of the graph offour units upward.

f

g

""!!1

!2

22

2

3

4

6

x

y

f#x$g#x$ " 4 # cos x

2$

f#x$ " cos x

28.

–2

2

x"6

fg

y

g#x$ " sin x3

f#x$ " sin x

30.

–2

2

x"

f

g

y

g#x$ " !cos 4x

f#x$ " 2 cos 2x

x 0

0 1 0 0

0 1"32

"32

12

sin x3

!1sin x

2$3$2

$$2

x 0

2 0 0 2

1 1 !1!1!1!cos 4x

!22 cos 2x

$3$4

$2

$4

31.

Period:

Amplitude:

is the graph of

shifted vertically three units upward.

f #x$g#x$ " 3 !12

sin x2

12

4$

x

y

"3"2 "4""4 !!

f

g

!2

!1

!3

!4

1

2

3

4

f#x$ " !12

sin x2

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Documents

crunchy math · CHAPTER 4 Trigonometric Functions Section 4.1 Radian and Degree Measure . . . . . . . . . . . . . . . . 272 Section 4.2 Trigonometric Functions: The Unit Circle