8/10/2019 Chapter 4 Trig Other Book
http://slidepdf.com/reader/full/chapter-4-trig-other-book 1/103
C H A P T E R 4
Trigonometry
Section 4.1 Radian and Degree Measure . . . . . . . . . . . . . . . . 335
Section 4.2 Trigonometric Functions: The Unit Circle . . . . . . . . . 344
Section 4.3 Right Triangle Trigonometry . . . . . . . . . . . . . . . . 350
Section 4.4 Trigonometric Functions of Any Angle . . . . . . . . . . 360
Section 4.5 Graphs of Sine and Cosine Functions . . . . . . . . . . . 373
Section 4.6 Graphs of Other Trigonometric Functions . . . . . . . . . 386
Section 4.7 Inverse Trigonometric Functions . . . . . . . . . . . . . . 397
Section 4.8 Applications and Models . . . . . . . . . . . . . . . . . . 410
Review Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 420
Problem Solving . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 432
Practice Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 436
8/10/2019 Chapter 4 Trig Other Book
http://slidepdf.com/reader/full/chapter-4-trig-other-book 2/103
335
C H A P T E R 4
Trigonometry
Section 4.1 Radian and Degree Measure
You should know the following basic facts about angles, their measurement, and their applications.
■ Types of Angles:
(a) Acute: Measure between 0 and 90.
(b) Right: Measure 90.
(c) Obtuse: Measure between 90 and 180 .
(d) Straight: Measure 180.
■ 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 1.
■ one second of 1.
■ The length of a circular arc is where is measured in radians.
■ Linear speed
■ Angular speed t srt
arc length
time
s
t
s r
160 of 1 136001
1601
180 .
1 180
180. 90.
Vocabulary Check
1. Trigonometry 2. angle
3. coterminal 4. radian
5. acute; obtuse 6. complementary; supplementary
7. degree 8. linear
9. angular 10. A 12r 2
1.
The angle shown is approximately
2 radians.
2.
The angle shown is approximately
5.5 radians.
3.
The angle shown is approximately
radians.3
4.
The angle shown is approximately
radians.4
5.
The angle shown is approximately
1 radian.
6.
The angle shown is approximately
6.5 radians.
8/10/2019 Chapter 4 Trig Other Book
http://slidepdf.com/reader/full/chapter-4-trig-other-book 3/103
336 Chapter 4 Trigonometry
7. (a) Since lies in Quadrant I.
(b) Since lies in Quadrant III. <7
5<
3
2 ;
7
5
0 <
5<
2 ;
5
9. (a) Since lies in Quadrant IV.
(b) Since lies in Quadrant III. < 2 <
2; 2
2 <
12 < 0 ;
12
8. (a) Since lies in Quadrant III.
(b) Since lies in Quadrant III. <9
8 <
3
2;
9
8
<11
8 <
3
2;
11
8
10. (a) Since lies in Quadrant IV.
(b) Since lies in Quadrant II.11
9
3
2 <
11
9 < ,
2< 1 < 0; 1
11. (a) Since lies in Quadrant III.
(b) Since lies in Quadrant II.
2< 2.25 < ; 2.25
< 3.5 <
3
2 ; 3.5 12. (a) Since lies in Quadrant IV.
(b) Since 4.25 lies in Quadrant II.3
2 < 4.25 < ;
3
2< 6.02 < 2 ; 6.02
13. (a)
(b)
−2
3
π
x
y
2
3
5
4
π
x
y
5
4 14. (a)
(b)
x
y
5
2
π
5
2
−7
4
π
y
x
7
4 15. (a)
(b)
−3
x
y
3
11
6
π
x
y
11
6
16. (a) 4
4
x
y (b)
7π
x
y7
8/10/2019 Chapter 4 Trig Other Book
http://slidepdf.com/reader/full/chapter-4-trig-other-book 4/103
Section 4.1 Radian and Degree Measure 337
26.
The angle shown is approximately
120.
27.
The angle shown is approximately
60º.
28.
The angle shown is approximately
330.
17. (a) Coterminal angles for
(b) Coterminal angles for
5
6 2
7
6
5
6 2
17
6
5
6
6 2
11
6
6 2
13
6
6 18. (a)
(b)
11
6 2
23
6
11
6 2
6
7
6 2
5
6
7
6 2
19
619. (a) Coterminal angles for
(b) Coterminal angles for
12 2
23
12
12 2
25
12
12
2
3 2
4
3
2
3 2
8
3
2
3
20. (a)
(b)
2
15 2
32
15
2
15 2
28
15
9
4
4 7
4
9
4 2
4 21. (a) Complement:
Supplement:
(b) Complement: Not possible, is greater than
Supplement: 3
4
4
2.
3
4
3
2
3
2
3
6
22. (a) Complement:
Supplement:
(b) Complement: Not possible, is greater than
Supplement: 11
12
12
2.
11
12
12
11
12
2
12
5
12 23. (a) Complement:
Supplement:
(b) Complement: Not possible, 2 is greater than
Supplement: 2 1.14
2.
1 2.14
2 1 0.57
24. (a) Complement: Not possible, 3 is greater than
Supplement:
(b) Complement:
Supplement: 1.5 1.64
2 1.5 0.07
3 0.14
2. 25.
The angle shown is approximately .210º
8/10/2019 Chapter 4 Trig Other Book
http://slidepdf.com/reader/full/chapter-4-trig-other-book 5/103
338 Chapter 4 Trigonometry
31. (a) Since lies in Quadrant II.
(b) Since lies in Quadrant IV.270 < 285 < 360, 285
90 < 130 < 180, 130 32. (a) Since lies in Quadrant I.
(b) Since lies in
Quadrant III.
180 < 257 30 < 270, 257 30
0 < 8.3 < 90, 8.3
33. (a) Since
lies in Quadrant III.
(b) Since lies in
Quadrant I.
360 < 336 < 270, 336
180 < 132 50 < 90, 132 50 34. (a) Since lies in
Quadrant II.
(b) Since lies in Quadrant IV.90 < 3.4 < 0, 3.4
270 < 260 < 180, 260
35. (a)
(b)
150°
x
y
150
x
30°
y
30 36. (a)
(b)
−120
°
x
y
120
−270°
x
y
270
37. (a)
(b)
480°
x
y
480
405°
x
y
405
29.
The angle shown is approximately .165
30.
The angle shown is approximately 10.
38. (a)
(b)
−600°
x
y
600
−750°
x
y
750
8/10/2019 Chapter 4 Trig Other Book
http://slidepdf.com/reader/full/chapter-4-trig-other-book 6/103
Section 4.1 Radian and Degree Measure 339
54. (a)
(b) 34 15
34 15 180
408
11
6
11
6 180
330 55.
2.007 radians
115 115
180 56.
1.525 radians
87.4 87.4
180
57. 216.35 216.35
180 3.776 radians
59. 532 532
180 9.285 radians
58. radians48.27 48.27
180 0.842
60. 345 345
180 6.021 radians
39. (a) Coterminal angles for 45
45 360 405
45 360 315
(b) Coterminal angles for
36 360 324
36 360 396
36
40. (a)
(b)
420 360 60
420 720 300
120 360 240
120 360 480 41. (a) Coterminal angles for
(b) Coterminal angles for
180 360 540
180 360 180
180
240 360 120
240 360 600
240
42. (a)
(b)
230 360 130
230 360 590
420 360 60
420 720 300 43. (a) Complement: 90 18 72
Supplement:
(b) Complement: Not possible, 115 is greater than 90 .
Supplement: 1180 115 65
180 18 162
44. (a) Complement:
Supplement:
(b) Complement:
Supplement: 180 64 116
90 64 26
180 3 177
90 3 87 45. (a) Complement:
Supplement:
(b) Complement: Not possible, is greater than
Supplement: 180 150 30
90.150
180 79 101
90 79 11
46. (a) Complement: Not possible, is greater than
Supplement:
(b) Complement: Not possible, is greater than
Supplement: 180 170 10
90.170
180 130 50
90.130 47. (a)
(b) 150 150
180 5
6
30 30
180
6
48. (a)
(b) 120 120
180 2
3
315 315
180 7
4 49. (a)
(b) 240 240
180 4
3
20 20
180
9 50. (a)
(b) 144 144
180 4
5
270 270
180 3
2
51. (a)
(b)7
6
7
6 180
210
3
2
3
2 180
270 52. (a)
(b)
9
9180
20
7
12
7
12180
105 53. (a)
(b) 11
30
11
30 180
66
7
3
7
3 180
420
8/10/2019 Chapter 4 Trig Other Book
http://slidepdf.com/reader/full/chapter-4-trig-other-book 7/103
340 Chapter 4 Trigonometry
61. radian0.83 0.83
180 0.014 62. 0.54 0.54
180 0.009 radians
63.
7
7180
25.714 64.
5
11
5
11180
81.818 65.15
8
15
8 180
337.500
66.13
2
13
2 180
1170.000 67.
756.000
4.2 4.2 180
68. 4.8 4.8 180
864.000
69. 2 2180
114.592 70. 0.57 0.57180
32.659
71. (a)
(b) 128 30 128 3060 128.5
54 45 54 4560 54.75
73. (a)
(b) 330 25 330 25
3600 330.007
85 18 30 85 1860 303600 85.308
75. (a)
(b) 145.8 145 0.860 145 48
240.6 240 0.660 240 36
77. (a)
(b)
3 34 48
3 34 0.860
3.58 3 0.5860
2.5 2 30
72. (a)
(b) 2 0.2 2.2212 2 1260
245.167 245 0.16724510 245 1060
74. (a)
(b)
408.272
408 0.2667 0.0056
408 16 20 408 1660 20
3600
135 0.01 135.01
135 36 135 363600
76. (a)
(b) 0 27
0 27 0.45 0 0.4560
345 7 12
345 7 0.260
345.12 345 0.1260
78. (a)
(b)
0 47 11.4 0 47 11.4
0 47 0.1960
0.7865 0 0.786560
0 21 18 0 21 18
0 21 0.360
0.355 0 0.35560
79.
65 radians
6 5
s r 81.
327 4
47 radians
32 7
s r 80.
2910 radians
29 10
s r
82.
Because the angle represented is
clockwise, this angle is radians.45
60
75
4
5 radians
60 75
s r 83.
6
27
2
9 radians
6 27
s r 84. feet, feet
s
r
8
14
4
7 radians
s 8r 14
8/10/2019 Chapter 4 Trig Other Book
http://slidepdf.com/reader/full/chapter-4-trig-other-book 8/103
Section 4.1 Radian and Degree Measure 341
100. sr 24
5 4.8 radians 4.8180
275
91.
8.38 square inches
A 1
242 3
8
3 square inches
A 1
2r 2 92.
56.55 mm2
18 mm2
A 1
2r 2
1
2122 4
r 12 mm,
4 93.
12.27 square feet
A 1
22.52225
180
A 1
2r 2
94.
square miles 5.6421.56
12 A
1
21.42330
180 r 1.4 miles, 330 95.
miless r 40000.14782 591.3
8.46972 0.14782 radian
41 15 50 32 47 39
96.
s r 40000.1715 686.2 miles
0.1715 radian
47 37 18 37 47 36 9 49 42
r 4000 miles97.
s
r
450
6378 0.071 radian 4.04
98. kilometers
The difference in latitude is about radians 3.59.0.062716
0.062716
400
6378
400 6378
s r
r 3189 99. s
r
2.5
6
25
60
5
12 radian
85.
25
14.5
50
29 radians
25 14.5
s r 87. in radians
47.12 inches
s 15180
180 15 inches
s r , 86. kilometers,
kilometers
s
r
160
80 2 radians
s 160
r 80
88. feet,
9.42 feet
s r 9 3 3 feet
60
3r 9 89. in radians
s 31 3 meters
s r , 90. centimeters,
15.71 centimeters
s r 20 4 5 centimeters
4r 20
101. (a) 65 miles per hour feet per minute
The circumference of the tire is feet.
The number of revolutions per minute is
revolutions per minuter 5720
2.5 728.3
C 2.5
655280
60 5720 (b) The angular speed is
Angular speed radians per minute4576 radians
1 minute 4576
5720
2.5 2 4576 radians
t .
8/10/2019 Chapter 4 Trig Other Book
http://slidepdf.com/reader/full/chapter-4-trig-other-book 9/103
342 Chapter 4 Trigonometry
102. Linear velocity for either pulley: inches per minute
(a) Angular speed of motor pulley: radians per minute
Angular speed of the saw arbor: radians per minute
(b) Revolutions per minute of the saw arbor: revolutions per minute
1700
2 850
v
r
3400
2 1700
v
r
3400
1 3400
17002 3400
103. (a)
(b)
9869.84 feet per minute
31412
3 feet per minute
Linear speed 7.25
2 in. 1 ft
12 in.52002
1 minute
32,672.56 radians per minute
10,400 radians per minute
Angular speed 52002 radians
1 minute104. (a)
(b)
628.32 feet per minute
Linear speed 2525.13274 feet per minute
r
t 200 feet per minute
r 25 ft
25.13 radians per minute
8 radians per minute
4 rpm 42 radians per minute
105. (a)
Interval:
(b)
Interval: 2400 , 6000 centimeters per minute
62002 ≤ Linear speed ≤ 65002 centimeters per minute
400 , 1000 radians per minute
2002 ≤ Angular speed ≤ 5002 radians per minute
106.
A 1
2125
180 252 112 175 549.8 square inches
r 25 14 11
R 25
25125°
14r
A 1
2 R2 r 2 107.
1496.62 square meters
476.39 square meters
12352140 180140°
35
A 1
2r 2
108. (a) Arc length of larger sprocket in feet:
Therefore, the chain moves feet, as does the smaller rear sprocket. Thus, the angle of the
smaller sprocket is
and the arc length of the tire in feet is:
—CONTINUED—
14 feet
3 seconds
3600 seconds
1 hour
1 mile
5280 feet 10 miles per hour
Speed s
t
14 3
1 second
14
3 feet per second
s 4 14
12 14
3 feet
s r
s
r
2 3 feet
212 feet
4
r 2 inches 212 feet. 2 3
s 1
32
2
3 feet
s r
8/10/2019 Chapter 4 Trig Other Book
http://slidepdf.com/reader/full/chapter-4-trig-other-book 10/103
Section 4.1 Radian and Degree Measure 343
115. The arc length is increasing. In order for the angle to
remain constant as the radius r increases, the arc length
s must increase in proportion to r, as can be seen from
the formula s r .
116. The area of a circle is The circumference of a circle is
For a sector, Thus, for a sector. A
r r
2
1
2 r
2
C
s
r .
Cr
2 A
C 2 A
r
C 2 Ar 2 r
C 2 r . A r 2 ⇒ A
r 2.
108. —CONTINUED—
(b) Since the arc length of the tire is feet and the cyclist is pedaling at a rate of one revolution per second,
we have:
Distance
(c)
(d) The functions are both linear.
14
3 feet per second 1 mile
5280 feett seconds 7
7920 t miles
Distance Rate Time
14
3
feet
revolutions1 mile
5280 feetn revolutions 7
7920n miles
14 3
109. False. An angle measure of radians corresponds to
two complete revolutions from the initial to the terminal
side of an angle.
4
111. False. The terminal side of lies on the negative x -axis.1260
113. Increases, since the linear speed is proportional to the radius.
110. True. If and are coterminal angles, then
where n is an integer. The difference
between is n360 2 n. and
n360
112. (a) An angle is in standard position if its vertex is at the origin and its initial side is on the positive x -axis.
(b) A negative angle is generated by a clockwise rotation of the terminal side.
(c) Two angles in standard position with the same terminal sides are coterminal.
(d) An obtuse angle measures between 90° and 180°.
114. 1 radian
so one radian is much larger than one degree.
180
57.3,
117.4
4 2
4
4 2 2
2
4 2
8
2
2 118.
5 5
2 10
5
2 510
5
2 12
5
2 2 2
2
5 2
4
119. 22 62 4 36 40 4 10 2 10 120.
208 16 13 4 13
172 92 289 81
8/10/2019 Chapter 4 Trig Other Book
http://slidepdf.com/reader/full/chapter-4-trig-other-book 11/103
344 Chapter 4 Trigonometry
Section 4.2 Trigonometric Functions: The Unit Circle
■ You should know the definition of the trigonometric functions in terms of the unit circle. Let t be a real number and
the point on the unit circle corresponding to t .
■ The cosine and secant functions are even.
■ The other four trigonometric functions are odd.
■ Be able to evaluate the trigonometric functions with a calculator.
cott cot t tant tan t
csct csc t sint sin t
sect sec t cost cos t
cot t x
y , y 0tan t y
x , x 0
sec t 1
x , x 0cos t x
csc t 1
y, y 0sin t y
x , y
Vocabulary Check1. unit circle 2. periodic
3. period 4. odd; even
121.
Graph of shifted
to the right by two units
y x 5
432−2
3
2
1
−1
−
2
−3
x
y y = x 5
y = ( x − 2)5
f x x 25 122.
Vertical shift four units
downward
4
2
−2
−6
32−2−3 1
y = x 5
y = x
5
−
4
x
y f x x 5 4
123.
Graph of reflected
in x -axis and shifted
upward by two units
y x 5
321−2−3
6
5
4
3
1
−1
−2
−3
x
y
y = x 5
y = 2 − x 5
f x 2 x 5 124.
Reflection in the x -axis
and a horizontal shift
three units to the left
3
2
1
−1
−2
−3
21−3−4−5 x
y
y = x 5
y = −( x + 3)5 f x x 35
8/10/2019 Chapter 4 Trig Other Book
http://slidepdf.com/reader/full/chapter-4-trig-other-book 12/103
Section 4.2 Trigonometric Functions: The Unit Circle 345
7. corresponds to 3
2,
1
2.t 7
6 8. t
5
4, x , y 2
2,
2
2
10. t 5
3, x , y 1
2,
3
2 9. corresponds to 1
2,
3
2 .t 4
3
11. corresponds to 0,1.t 3 2
13. corresponds to
sin
cos
tan t y
x 1
t x 2
2
t y 2
2
2
2, 2
2 .t
4
15. corresponds to
sin
cos
tan t y
x
1
3
3
3
t x 3
2
t y 1
2
3
2,
1
2.t
6
12. t , x , y 1, 0
14.
tan
3
32
12 3
cos
3
1
2
sin
3
3
2
t
3, x , y 1
2, 3
2
16.
tan
4 22
22 1
cos
4 2
2
sin
4 2
2
t
4, x , y 2
2,
2
2
1.
sin csc
cos sec
tan cot x
y
8
15
y
x
15
8
1
x
17
8 x
8
17
1
y
17
15 y
15
17
x 8
17, y
15
17
3.
sin csc
cos sec
tan cot x
y
12
5
y
x
5
12
1
x
13
12 x
12
13
1
y
13
5 y
5
13
x 12
13, y
5
13
5. corresponds to 2
2, 2
2 .t
4
2.
cot x
y
12
5tan
y
x
5
12
sec 1
x
13
12cos x
12
13
csc 1
y
13
5sin y
5
13
x 12
13, y
5
13
4.
cot x
y
4
3tan
y
x
3
4
sec 1
x
5
4cos x
4
5
csc 1
y
5
3sin y
3
5
x 4
5, y
3
5
6. t
3, x y 1
2, 3
2
8/10/2019 Chapter 4 Trig Other Book
http://slidepdf.com/reader/full/chapter-4-trig-other-book 13/103
346 Chapter 4 Trigonometry
19. corresponds to
sin
cos
tan t y
x
1
3
3
3
t x 3
2
t y 1
2
3
2,
1
2.t 11
620.
tan5
3
32
12 3
cos5
3
1
2
sin5
3
3
2
t 5
3, x , y 1
2,
3
2
21. corresponds to
sin
cos
tan is undefined.t y
x
t x 0
t y 1
0, 1.t 3
222.
tan2 0
1 0
cos2 1
sin2 0
t 2 , x , y 1, 0
23. corresponds to
sin csc
cos sec
tan cot t x
y 1t
y
x 1
t 1
x 2t x
2
2
t 1
y 2t y
2
2
2
2, 2
2 .t 3
424.
cot5
6
1
tan t 3tan
5
6
12
32
3
3
sec5
6
1
cos t
2 3
3cos
5
6
3
2
csc5
6
1
sin t 2sin
5
6
1
2
t 5
6, x , y 3
2,
1
2
25. corresponds to
sin csc
cos sec is undefined.
tan is undefined. cot t x
y 0t
y
x
t 1
x t x 0
t 1
y
1t y 1
0,1.t
226.
is undefined.
is undefined. cot3
2
0
1 0tan
3
2
sec3
2cos
3
2 0
csc3
2
1
sin t
1sin3
2
1
t 3
2, x , y 0,1
17. corresponds to
sin
cos
tan t y
x 1
t x 2
2
t y 2
2
2
2, 2
2 .t 7
4 18.
tan4
3 3212
3
cos4
3 1
2
sin4
3 3
2
t 4
3, x , y 1
2, 3
2
8/10/2019 Chapter 4 Trig Other Book
http://slidepdf.com/reader/full/chapter-4-trig-other-book 14/103
Section 4.2 Trigonometric Functions: The Unit Circle 347
29. sin 5 sin 0 31. cos8
3 cos
2
3
1
2
33. cos15
2 cos
2 0
35. sin9
4 sin7
4 2
237.
(a)
(b) csct csc t 3
sint sin t 1
3
sin t 1
3
39.
(a)
(b) sect 1
cost 5
cos t cost 1
5
cost 1
5
30. cos 5 cos 1
32. sin9
4 sin
4
2
2 34. sin
19
6 sin
7
6
1
2
36. cos8
3 cos4
3
1
2
38.
(a)
(b) csc t 1
sin t
8
3
sin t sint 3
8
sint 3
840.
(a)
(b) sect sec t 1
cos t
4
3
cost cos t 3
4
cos t 3
4
41.
(a)
(b) sint sin t 4
5
sin t sin t 4
5
sin t 4
543. sin
4 0.7071
45. csc 1.3 1
sin 1.3 1.0378
42.
(a)
(b) cost cos t 4
5
cos t cos t 4
5
cos t 4
5
44. tan
3 1.7321 46. cot 1
1
tan 1 0.6421
47. cos1.7 0.1288 49. csc 0.8 1
sin 0.8 1.3940
51. sec 22.8 1
cos 22.8 1.4486
53. (a)
(b) cos 2 x 0.4
sin 5 y 1
48. cos2.5 0.8011
50. sec 1.8 1
0051.8 4.4014 52. sin0.9 0.7833
54. (a)
(b) cos 2.5 x 0.8
sin 0.75 y 0.7
27. corresponds to
sin csc
cos sec
tan cot t x
y 3
3t
y
x 3
t 1
x 2t x
1
2
t 1
y
2 3
3t y
3
2
1
2,
3
2 .t 4
328.
cot7
4
1
tan t 1tan
7
4
22
22 1
sec7
4
1
cos t 2cos
7
4
2
2
csc7
4
1
sin t 2sin
7
4
2
2
t 7
4, x , y 2
2,
2
2
8/10/2019 Chapter 4 Trig Other Book
http://slidepdf.com/reader/full/chapter-4-trig-other-book 15/103
348 Chapter 4 Trigonometry
57.
(a)
(b) From the table feature of a graphing utility we see that seconds.
(c) As t increases, the displacement oscillates but decreases in amplitude.
y 0 when t 5
yt 1
4 et
cos 6t
t 0 1
y 0.25 0.0138 0.08830.02490.1501
34
12
14
58.
(a)
(b)
(c) y1
2 1
4 cos 3 0.2475 feet
y1
4 1
4 cos
3
2 0.0177 feet
y0 1
4 cos 0 0.2500 feet
yt 1
4 cos 6t 59. False. means the function is odd, not
that the sine of a negative angle is a negative number.
For example:
Even though the angle is negative, the sine value is positive.
sin
3
2 sin
3
2 1 1.
sint sin t
60. True. since the period of tan is .tan a tana 6 61. (a) The points have y-axis symmetry.
(b) since they have the same y-value.
(c) since the x -values have the
opposite signs.
cos t 1 cos t
1
sin t 1 sin t
1
62.
So and are even.
So and are odd.
So and are odd.cot tan
cot x
y cot
cot x
y
tan y
x tan
tan y
x
csc sin
csc 1
y csc
x
y
θ
θ −
( x , y)
( x , − y)
csc 1
y
sin y sin
sin y
cos sec
sec 1
x sec
cos x cos 63.
f 1 x 2
3 x
2
3
2
3 x 1
2
3 x
2
3 y
2 x 3 y 2
x 1
23 y 2
y 1
23 x 2
f x 1
23 x 2
55. (a)
(b)
t 1.82 or 4.46
cos t 0.25
t 0.25 or 2.89
sin t 0.25 56. (a)
(b)
t 0.7 or t 5.6
cos t 0.75
t 4.0 or t 5.4
sin t 0.75
8/10/2019 Chapter 4 Trig Other Book
http://slidepdf.com/reader/full/chapter-4-trig-other-book 16/103
Section 4.2 Trigonometric Functions: The Unit Circle 349
64.
f 1 x 3 4 x 1
y 3 4 x 1
4 x 1 y3
x 1 1
4 y3
x 1
4 y3 1
y 1
4 x 3 1
f x 1
4 x 3 1
66.
f 1 x 22 x 1
x 1
y 22 x 1
x 1
y x 1 4 x 2
xy y 4 x 2
xy 4 x y 2
x y 4 y 2
x y 2
y 4
y x 2
x 4
f x x 2
x 465.
f 1 x x 2 4, x ≥ 0
± x 2 4 y
x 2 y2 4
x y2 4
y x 2 4
f x x 2 4, x ≥ 2
67.
Intercept:
Vertical asymptote:
Horizontal asymptote: y 2
x 3
0, 0
y
x −2−4−6 2 4 6 8 10−2
−4
−6
−8
4
6
8
2
f x 2 x
x
3
x 0 1 2 4 5 6
y 0 8 5 4411
2
1
68.
Horizontal asymptote:
Vertical asymptote:
Intercept: 0, 0
x 3, x 2
x 0
y
x −4 4 6 8
−2
−4
−6
−8
2
4
6
8
2−2
f x 5 x
x 2
x 6
5 x
x 3 x 2, x 3, 2
x 0 1 3 5
y 025
24
5
2
5
4
5
2
10
3
5
4
246
69.
Intercepts:
Vertical asymptote:
Horizontal asymptote:
Hole in the graph at 2,7
8
y 1
2
x 2
5, 0, 0,5
4
y
x −1−5−6 1 2−1
−2
−3
−4
1
2
3
4
−2
f x x 2 3 x 10
2 x 2 8
x 5 x 2
2 x 2 x 2 x 5
2 x 2, x 2
x 0 1 3
y 0 2 14
5
5
41
1
4
1345
8/10/2019 Chapter 4 Trig Other Book
http://slidepdf.com/reader/full/chapter-4-trig-other-book 17/103
350 Chapter 4 Trigonometry
70.
Vertical asymptote:
Slant asymptote:
intercept:
intercept: 5.86, 0 x -
0,1
8 y-
y 1
2 x
7
4
x
5 ± 89
4 ; x
1.11, x 3.61
y
x −4 46 82−2−6
−2
x 5 ± 52 428
22
2 x 2 5 x 8 0
f x x 3 6 x 2 x 1
2 x 2 5 x 8
1
2 x
7
4
15 x 4
42 x 2 5 x 8
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 that cofunctions of complemen-
tary angles are equal.
■ You should know the trigonometric function values of 30 , 45 , and 60 , or be able to construct triangles from which you
can determine them.
90, and
1 cot2 csc2 1 tan2 sec2
sin2
cos2
1cot cos
sin tan sin
cos
cot 1
tan tan
1
cot sec
1
cos
cos 1
sec csc
1
sin sin
1
csc
cot adj
oppsec
hyp
adjcsc
hyp
opp
tan opp
adjcos
adj
hypsin
opp
hyp
0 2 3 4 7
9 5 1294
32
18
15532
175
154 y
13234 x
Vocabulary Check
1. (i) (v) (ii) (iv) (iii) (vi)
(iv) (iii) (v) (i) (vi) (ii)
2. opposite; adjacent; hypotenuse
3. elevation; depression
opposite
adjacent tan
opposite
hypotenuse sin
adjacent
hypotenuse cos
hypotenuseopposite
csc adjacentopposite
cot hypotenuseadjacent
sec
8/10/2019 Chapter 4 Trig Other Book
http://slidepdf.com/reader/full/chapter-4-trig-other-book 18/103
Section 4.3 Right Triangle Trigonometry 351
5.
tan opp
adj
1
2
2
2
4
cos adj
hyp
2 2
3
sin opp
hyp
1
3θ
31
adj 32 12 8 2 2
cot adj
opp
2 2
sec hyp
adj
3
2 2
3 2
4
csc hyp
opp 3
tan opp
adj
2
4 2
1
2 2
2
4
cos adj
hyp
4 2
6
2 2
3
sin opp
hyp
2
6
1
3θ
62
adj 62 22 32 4 2
cot adj
opp
4 2
2 2 2
sec hyp
adj
6
4 2
3
2 2
3 2
4
csc hyp
opp
6
2 3
The function values are the same since the triangles are similar and the corresponding sides are proportional.
1.
tan opp
adj
6
8
3
4
cos adj
hyp
8
10
4
5
sin opp
hyp
6
10
3
5
6
8
θ
hyp 62 82 36 64 100 10
cot adj
opp
8
6
4
3
sec hyp
adj
10
8
5
4
csc hyp
opp
10
6
5
3
2.
513
b
θ sin csc
cos sec
tan cot adj
opp
12
5
opp
adj
5
12
hyp
adj
13
12
adj
hyp
12
13
hyp
opp
13
5
opp
hyp
5
13
adj 132 52 169 25 12
3.
tan opp
adj
9
40
cos adj
hyp
40
41
sin opp
hyp
9
41θ 9
41
adj 412 92 1681 81 1600 40
cot adj
opp
40
9
sec hyp
adj
41
40
csc hyp
opp
41
9
4.
4
4
θ
sin csc
cos sec
tan cot adj
opp
4
4 1
opp
adj
4
4 1
hypadj
4 24 2 adj
hyp 4
4 2 1 2
22
hyp
opp
4 2
4 2
opp
hyp
4
4 2
1
2 2
2
hyp 42 42 32 4 2
8/10/2019 Chapter 4 Trig Other Book
http://slidepdf.com/reader/full/chapter-4-trig-other-book 19/103
352 Chapter 4 Trigonometry
6.
cot adj
opp
15
8tan
opp
adj
8
15
sec hyp
adj
17
15cos
adj
hyp
15
17
csc hypopp
178
sin opphyp
8
17
hyp 152 82 289 17
15
8
θ
cot adj
opp
7.5
4
15
8tan
opp
adj
4
7.5
8
15
sec hyp
adj
172
7.5
17
15cos
adj
hyp
7.5
172
15
17
csc hyp
opp
172
4
17
8
sin opp
hyp
4
172
8
17
hyp 7.52 42 17
2
7.5
4
θ
The function values are the same because the triangles are similar, and corresponding sides are proportional.
7.
tan opp
adj
3
4
cos adj
hyp
4
5
sin opp
hyp
3
5
θ
4
5
opp 52 42 3
cot adj
opp
4
3
sec hyp
adj
5
4
csc hyp
opp
5
3
The function values are the same since the triangles are similar and the corresponding sides are proportional.
tan
opp
adj
0.75
1
3
4
cos adj
hyp
1
1.25
4
5
sin opp
hyp
0.75
1.25
3
5θ
1
1.25
opp 1.252 12 0.75
cot adj
opp 1
0.75
4
3
sec hyp
adj
1.25
1
5
4
csc hyp
opp
1.25
0.75
5
3
8.
cot adj
opp
2
1 2tan
opp
adj
1
2
sec hyp
adj 5
2cos
adj
hyp
2
5
2 5
5
csc hyp
opp 5
1 5sin
opp
hyp
1
5 5
5
hyp 12 22 5
1
2
θ
cot 6
3 2tan
3
6
1
2
sec 3 5
6 5
2cos
6
3 5
2
5
2 5
5
csc 3 5
3 5sin
3
3 5
1
5 5
5
hyp 32 62 3 5
3
6
θ
The function values are the same because the triangles are similar, and corresponding sides are proportional.
8/10/2019 Chapter 4 Trig Other Book
http://slidepdf.com/reader/full/chapter-4-trig-other-book 20/103
Section 4.3 Right Triangle Trigonometry 353
13. Given:
cot adj
opp
1
3
sec hyp
adj 10
csc hyp
opp
10
3
cos adj
hyp 10
10
sin opp
hyp
3 10
10
hyp 10
32 12 hyp2
3
1
10
θ
tan 3 3
1
opp
adj14. Given:
cot adj
opp
1
35 35
35
csc hyp
opp
6
35
6 35
35
tan opp
adj
35
1 35
cos adj
hyp
1
6
sin opp
hyp 35
6
opp 62 12 35
6
1
θ
35
sec 6
1
hyp
adj
9. Given:
cot adj
opp 7
3
sec hyp
adj
4 7
7
csc hyp
opp
4
3
tan opp
adj
3 7
7
cos adj
hyp 7
4
adj 7
32 adj2 42
θ
7
43
sin 3
4
opp
hyp
11. Given:
cot adj
opp 3
3
csc hyp
opp
2 3
3
tan opp
adj 3
cos adj
hyp
1
2
sin opp
hyp 3
2
opp 3
1
23
θ
opp2
12
22
sec 2 2
1
hyp
adj
10. Given:
cot adj
opp
5
2 6
5 6
12
sec hyp
adj
7
5
csc hyp
opp
7
2 6
7 6
12
tan
opp
adj
2 6
5
sin opp
hyp
2 6
7
opp 72 52 24 2 6
5
72 6
θ
cos 5
7
adj
hyp
12. Given:
sec hyp
adj
26
5
csc hyp
opp 26
1 26
tan opp
adj
1
5
cos adj
hyp
5
26
5 26
26
sin opp
hyp
1
26 26
26
hyp 52 12 26
261
5
θ cot 5
1
adj
opp
8/10/2019 Chapter 4 Trig Other Book
http://slidepdf.com/reader/full/chapter-4-trig-other-book 21/103
354 Chapter 4 Trigonometry
15. Given:
sec hyp
adj 13
3
csc hyp
opp 13
2
tan opp
adj
2
3
cos adj
hyp
3
13
3 13
13
sin opp
hyp
2
13
2 13
13
hyp 13
22 32 hyp2 2
3
13
θ
cot 3
2
adj
opp16. Given:
cot adj
opp 273
4
sec hyp
adj
17
273
17 273
273
tan opp
adj
4
273
4 273
273
cos
adj
hyp
273
17
sin opp
hyp
4
17
adj 172 42 273
417
273
θ
csc 17
4
hyp
opp
17.
sin 30 opp
hyp
1
2
30 30 180
6 radian
30°
60°
1
2
3
18.
cos 45 adj
hyp
1
2 2
2
45 45 180
4 radian
2
1
1
45°
19.
tan
3
opp
adj
3
1 3
3
3180
60
1
23
π
3
π
6
20.
sec
4
hyp
adj
2
1 2
4
4180
45
2
1
1
π
4
21.
60
3 radian
cot 3
3
1
3
adj
opp
1
3
30°
60°
22.
45 45 180
4 radian
csc 2 hyp
opp
2
1
1
45°
23.
cos 6 adj
hyp 3
2
6
6180
30
12
3
π
3
π
6
24.
sin 4 opp
hyp 1 2
22
4
4180
45
2
1
1
π
4
25.
4 radian
45 45 180
cot 1 1
1
adj
opp2
1
1
45°
45°
26.
6 radian
30 30 180
tan 3
3
1
3
opp
adj
30°
60°1
2
3
8/10/2019 Chapter 4 Trig Other Book
http://slidepdf.com/reader/full/chapter-4-trig-other-book 22/103
Section 4.3 Right Triangle Trigonometry 355
31.
(a)
(b)
(c)
(d) sin90 cos 1
3
cot cos sin
1
3
2 2
3
1
2 2 2
4
sin 2 2
3
sin2 8
9
sin2 1
32
1
sin2 cos2 1
sec 1cos
3
cos 1
3 32.
(a)
(b)
(c)
(d)
1 125
2625
265
1 1
52
csc 1 cot2
tan90º cot 1
tan
1
5
1
1 52
1
26 26
26
cos 1
sec
1
1 tan2
cot 1
tan 1
5
tan 5
33. tan cot tan 1
tan 1 34. cos sec cos 1
cos 1
27.
(a)
(b)
(c)
(d) cot 60 cos 60
sin 60
1
3 3
3
cos 30 sin 60 32
sin 30 cos 60 1
2
tan 60 sin 60
cos 60 3
sin 60 3
2, cos 60
1
2 28.
(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 tan90 60 tan 30 3
3
csc 30 1
sin 30 2
sin 30 1
2, tan 30
3
3
29.
(a)
(b)
(c)
(d) sec90 csc 13
2
tan sin
cos
2 13
13
3 13
13
2
3
cos
1
sec
3
13
3 13
13
sin 1
csc
2
13
2 13
13
csc 13
2, sec
13
3 30.
(a)
(b)
(c)
(d) sin tan cos 2 6 1
5 2 6
5
cot90º tan 2 6
cot 1
tan
1
2 6 6
12
cos 1
sec
1
5
sec 5, tan 2 6
8/10/2019 Chapter 4 Trig Other Book
http://slidepdf.com/reader/full/chapter-4-trig-other-book 23/103
356 Chapter 4 Trigonometry
35. tan cos sin
cos cos sin 36. cot sin cos
sin sin cos
37.
sin
2
sin2 cos2 cos2
1 cos 1 cos 1 cos2 38. 1 sin 1 sin 1 sin2 cos2
39.
1
1 tan2 tan2
sec tan sec tan sec2 tan2
41.
csc sec
1
sin
1
cos
1
sin cos
sin
cos
cos
sin
sin2 cos2
sin cos
40.
2 sin2 1
sin2 1 sin2
sin2 cos2 sin2 1 sin2
42.
1 cot2 csc2
1 cot
1
cot
tan cot
tan
tan
tan
cot
tan
43. (a)
(b)
Note: cos 80 sin90 80 sin 10
cos 80 0.1736
sin 10 0.1736 44. (a)
(b) cot 66.5 1
tan 66.5 0.4348
tan 23.5 0.4348
45. (a)
(b) csc 16.35
1
sin 16.35 3.5523
sin 16.35 0.2815 46. (a)
(b) sin 73 56 sin73 56
60
0.9609
cos 16 18 cos16 18
60 0.9598
47. (a)
(b) csc 48 7 1
sin 48 760 1.3432
sec 42 12 sec 42.2 1
cos 42.2 1.3499
49. (a)
(b) tan 11 15 tan 11.25 0.1989
cot 11 15
1
tan 11.25
5.0273
51. (a)
(b) tan 44 28 16 tan 44.4711 0.9817
csc 32 40 3 1
sin 32.6675 1.8527
48. (a)
(b)
1.0036
sec 4 50 15 1
cos 4 50 15
0.9964
cos 4 50 15 cos4 50
60
15
3600
50. (a)
(b) cos 56 8 10 cos56 8
60
10
3600 0.5572
sec 56 8 10
sec56
8
60
10
3600
1.7946
52. (a)
(b) cot9
5 30 32 0.0699
sec9
5 20 32 2.6695
8/10/2019 Chapter 4 Trig Other Book
http://slidepdf.com/reader/full/chapter-4-trig-other-book 24/103
Section 4.3 Right Triangle Trigonometry 357
62.
r 20
sin 45
20
22 20 2
sin 45 20
r
63.
Height of the building:
Distance between friends:
323.34 meters
cos 82 45
y ⇒ y
45
cos 82
123 45 tan 82 443.2 meters
x 45 tan 82
45 m
82°
x y
tan 82 x
45
53. (a)
(b) csc 2 ⇒ 30
6
sin 1
2 ⇒ 30
655. (a)
(b) cot 1 ⇒ 45
4
sec 2 ⇒ 60
354. (a)
(b) tan 1 ⇒ 45º
4
cos 2
2 ⇒ 45
4
59.
x 30 3
1 3 30 x
tan 30 30
x
30°
30
x
60.
y 18 sin 60 18
3
2 9 3
sin 60 y
18
61.
x 32
3
32 3
3
3 x 32
3 32
x
tan 60 32
x
60°
32
x
64. (a) (b)
(c)
h 270 feet
2135 h
tan 6
3
h
135
h
3
6
132
Not drawn to scale
65.
0
6
sin 1500
3000
1
2
θ
1500 ft3000 ft
66.
w 100 tan 54 137.6 feet
tan 54 w
100
tan opp
adj
56. (a)
(b) cos 1
2 ⇒ 60
3
tan 3 ⇒ 60
3
57. (a)
(b) sin 2
2 ⇒ 45
4
csc
2 3
3 ⇒ 60
3
58. (a)
tan 3
3 3 ⇒ 60
3
cot 3
3 (b)
cos 1
2 2
2 ⇒ 45
4
sec 2
8/10/2019 Chapter 4 Trig Other Book
http://slidepdf.com/reader/full/chapter-4-trig-other-book 25/103
358 Chapter 4 Trigonometry
67.
(a)
x 145sin 23
371.1 feet
sin 23 145
x
150 ft
5 ft23°
x
(b)
y 145tan 23
341.6 feet
tan 23 145
y(c) Moving down the line:
feet per second
Dropping vertically:
feet per second145
6 24.17
145sin 23
6 61.85
68. Let the height of the mountain.
Let the horizontal distance from where the angle of
elevation is sighted to the point at that level directly below
the mountain peak.
Then tan
Substitute into the expression for tan
The mountain is about 1.3 miles high.
1.2953 h
13 tan 9 tan 3.5tan 9 tan 3.5
h
13 tan 9 tan 3.5 htan 9 tan 3.5
h tan 3.5 13 tan 9 tan 3.5 h tan 9
tan 3.5 h tan 9
h 13 tan 9
tan 3.5 h
h
tan 9 13
3.5. x h
tan 9
tan 9 h
x ⇒ x
h
tan 9
3.5 h
x 13 and tan 9
h
x .
9 x
h 69.
x 2, y
2 28, 28 3
x 2 cos 60°56 1
256 28
cos 60° x
2
56
y2 sin 60°56 3
2 56 28 3
sin 60 y
2
56
60°
56
( , ) x y2 2
x 1, y
1 28 3, 28
x 1 cos 3056
3
2 56 28 3
cos 30 x
1
56
y1 sin 3056 1
256 28
sin 30
y1
56
30°
56
( , ) x y1 1
70.
d 5 2 x 5 215 tan 3 6.57 centimeters
x 15 tan 3
tan 3 x
15
8/10/2019 Chapter 4 Trig Other Book
http://slidepdf.com/reader/full/chapter-4-trig-other-book 26/103
Section 4.3 Right Triangle Trigonometry 359
72.
tan 20 y
x 0.36
cos 20 x
10
0.94
sin 20 y
10 0.34
x 9.4, y 3.4
csc 20 10
y 2.92
sec 20 10
x
1.06
cot 20 x
y 2.75
73. True,
csc x 1
sin x ⇒ sin 60 csc 60 sin 60 1
sin 60 1
75. False, 2
2 2
2 2 1
77. False,
sin 2 0.0349
sin 60
sin 30
cos 30
sin 30 cot 30 1.7321;
74. True, because sec90 csc .sec 30 csc 60
76. True, because
cot2 csc2 1.
cot2 csc2 1
1 cot2 csc2
cot2 10 csc2 10 1
78. False,
tan25 tan 5tan 5 0.0077
tan52 tan 25 0.4663
tan52 tan25. 79. This is true because the corresponding sides of similar
triangles are proportional.
80. Yes. Given can be found from the identity 1 tan2 sec2 .tan , sec
81. (a) (b) In the interval
(c) As approaches 0, approaches .sin
0, 0.5, > sin .0.1 0.2 0.3 0.4 0.5
0.0998 0.1987 0.2955 0.3894 0.4794sin
71. (a)
(b)
(c)
(d) The side of the triangle labeled
h will become shorter.
h 20 sin 85 19.9 meters
sin 85 h20
h20
85°
(e)Angle, Height (in meters)
80 19.7
70 18.8
60 17.3
50 15.3
40 12.9
30 10.0
20 6.8
10 3.5
(f) The height of the balloon
decreases as decreases.
20 h
θ
82. (a)
—CONTINUED—
sin 0 0.3090 0.5878 0.8090 0.9511 1
cos 1 0.9511 0.8090 0.5878 0.3090 0
90725436180
8/10/2019 Chapter 4 Trig Other Book
http://slidepdf.com/reader/full/chapter-4-trig-other-book 27/103
8/10/2019 Chapter 4 Trig Other Book
http://slidepdf.com/reader/full/chapter-4-trig-other-book 28/103
Section 4.4 Trigonometric Functions of Any Angle 361
Vocabulary Check
1. 2. 3.
4. 5. 6.
7. reference
cot x
r cos
r
x
tan y
x csc sin
y
r
1. (a)
r 16 9 5
x , y 4, 3 (b)
r 64 225 17
x , y 8, 15
tan y
x
3
4
cos x
r
4
5
sin y
r
3
5
cot x
y
4
3
sec r
x
5
4
csc r
y
5
3
tan y
x
15
8
cos x
r
8
17
sin y
r
15
17
cot x
y
8
15
sec r
x
17
8
csc r
y
17
15
2. (a)
cot x
y12
5
12
5
sec r
x
13
12
13
12
csc r
y
13
5
13
5
tan y
x 5
12
5
12
cos x
r
12
13
sin y
r
5
13
r 122 52 13
x 12, y 5 (b)
cot x
y1
1 1
sec r
x 2
1 2
csc r
y
2
1
2
tan y
x
1
1 1
cos x
r 1
2
2
2
sin y
r
1
2 2
2
r 12 12 2
x 1, y 1
3. (a)
r 3 1 2
x , y 3, 1 (b)
r 16 1 17
x , y 4, 1
tan y
x
3
3
cos x r 3
2
sin y
r
1
2
cot x
y 3
sec r
x
2 33
csc r
y 2
tan y
x
1
4
cos x r 4 17
17
sin y
r 17
17
cot x
y 4
sec r x 17
4
csc r
y 17
8/10/2019 Chapter 4 Trig Other Book
http://slidepdf.com/reader/full/chapter-4-trig-other-book 29/103
362 Chapter 4 Trigonometry
4. (a)
cot x
y
3
1 3
sec r
x 10
3
csc r
y 10
1 10
tan y
x
1
3
cos x
r
3
10
3 10
10
sin y
r
1
10 10
10
r 32 12 10
x 3, y 1 (b)
cot x
y
4
4 1
sec r
x
4 2
4 2
csc r
y
4 2
4 2
tan y
x 4
4 1
cos x
r
4
4 2 2
2
sin y
r 4
4 2
2
2
r 42 42 4 2
x 4, y 4
5.
cot x
y
7
24
sec r
x
25
7
csc r
y
25
24
tan y
x
24
7
cos x
r
7
25
sin y
r
24
25
r 49 576 25
x , y 7, 24 6.
cot x
y
8
15
sec r
x
17
8
csc r
y
17
15
tan y
x
15
8
cos x
r
8
17
sin y
r
15
17
r 82 152 17
x 8, y 15 7.
cot x
y
2
5
sec r
x
29
2
csc r
y 29
5
tan y
x
5
2
cos x
r
2 29
29
sin y
r
5 29
29
r 16 100 2 29
x , y 4, 10
8.
cot x
y5
2
5
2
sec r
x 29
5
29
5
csc r
y 29
2
29
2
tan y
x 2
5
2
5
cos x
r 5
29
5 29
29
sin y
r 2
29
2 29
29
r 52 22 29
x 5, y 2 9.
cot x
y
35
68 0.5
sec r
x
5849
35 2.2
csc r
y 5849
68 1.1
tan y
x
68
35 1.9
cos x
r
35 5849
5849 0.5
sin y
r
68 5849
5849 0.9
r 12.25 46.24 5849
10
x , y 3.5, 6.8
8/10/2019 Chapter 4 Trig Other Book
http://slidepdf.com/reader/full/chapter-4-trig-other-book 30/103
Section 4.4 Trigonometric Functions of Any Angle 363
18.
cot
8
15tan
y
x 15
8
15
8
sec 17
8cos
x
r
8
17
csc 17
15sin
y
r 15
17
15
17
tan < 0 ⇒ y 15
cos x
r
8
17 ⇒ y 15
15.
tan y
x
3
4
cos x
r
4
5
sin y
r
3
5
in Quadrant II ⇒ x 4
sin y
r
3
5 ⇒ x 2 25 9 16
10.
cot x
y
72
314
14
31 0.5tan
y
x 314
72
31
14 2.2
sec r x 11574
72 1157
14 2.4cos
x r
72 11574
14 1157
1157 0.4
csc r
y 11574
314
1157
31 1.1sin
y
r
314
11574
31 1157
1157 0.9
r 7
22
31
4 2
1157
4
x 31
2
7
2, y 7
3
4
31
4
11.
sin < 0 and cos < 0 ⇒ lies in Quadrant III.
cos < 0 ⇒ lies in Quadrant II or in Quadrant III.
sin < 0 ⇒ lies in Quadrant III or in Quadrant IV.
13.
sin > 0 and tan < 0 ⇒ lies in Quadrant II.
tan < 0 ⇒ lies in Quadrant II or in Quadrant IV.
sin > 0 ⇒ lies in Quadrant I or in Quadrant II.
12. and
and
Quadrant I
x
r > 0
y
r > 0
cos > 0sin > 0
14. and
and
Quadrant IV
x
y< 0
r
x > 0
cot < 0sec > 0
cot x
y
4
3
sec r
x
5
4
csc r
y
5
3
16.
in Quadrant III
tan y
x
3
4
cos x
r
4
5
sin y
r
3
5
⇒ y 3
cos x
r 4
5 ⇒ y2 25 16 9
cot 4
3
sec 5
4
csc 5
3
17.
is in Quadrant IV
tan y
x
15
8
cos
x
r
8
17
sin y
r
15
17
x 8, y 15, r 17
y < 0 and x > 0.
⇒sin < 0 and tan < 0 ⇒
tan y
x 15
8
cot x
y
8
15
sec
r
x
17
8
csc r
y
17
15
8/10/2019 Chapter 4 Trig Other Book
http://slidepdf.com/reader/full/chapter-4-trig-other-book 31/103
364 Chapter 4 Trigonometry
19.
cot x
y 3tan
y
x
1
3
sec r
x 10
3cos
x
r
3 10
10
csc r
y 10sin
y
r
10
10
x 3, y 1, r 10
cos > 0 ⇒ is in Quadrant IV ⇒ x is positive;
cot x
y
3
1
3
120.
cot 15tan y
x
15
15
sec
4 15
15cos
x
r
15
4
csc 4sin y
r
1
4
cot < 0 ⇒ x 15
csc r
y
4
1 ⇒ x 15
21.
cot x
y
3
3tan
y
x 3
sec
r
x 2cos
x
r
1
2
csc r
y
2 3
3sin
y
r 3
2
sin > 0 ⇒ is in Quadrant II ⇒ y 3
sec r
x
2
1 ⇒ y2 4 1 3 22.
is undefined
is undefinedcot x
ytan
y
x 0
sec r x 1cos x
r r
r 1
csc r
ysin 0
y 0, x r
sec 1 ⇒ 2 n
sin 0 ⇒ 0 n
23.
cot is undefinedtan 0
sec 1cos 1
csc is undefinedsin 0
cot is undefined,
2 ≤ ≤
3
2 ⇒ y 0 ⇒ 24. tan is undefined
is undefined.
is undefined. cot x
y
0
y 0tan
y
x
sec
r
x cos
x
r
0
r
0
csc r
y 1sin
y
r r
r 1
≤ ≤ 2 ⇒ 3
2, x 0, y r
⇒ n
2
25. To find a point on the terminal side of use any point on
the line that lies in Quadrant II. is one
such point.
cot 1
sec 2
csc 2
tan 1
cos
1
2
2
2
sin 1
2 2
2
x 1, y 1, r 2
1, 1 y x
26. Let
Quadrant III
cot x
y
x
13 x 3
sec r
x
10 x 3
x
10
3
csc r
y
10 x 3
13 x 10
tan y
x
13 x
x
1
3
cos x r x
10 x 3 3 10
10
sin y
r
13 x
10 x 3
10
10
r x 2 1
9 x 2
10 x
3
x , 1
3 x ,
x > 0.
8/10/2019 Chapter 4 Trig Other Book
http://slidepdf.com/reader/full/chapter-4-trig-other-book 32/103
Section 4.4 Trigonometric Functions of Any Angle 365
27. use any point on
the line Quadrant III. is one
such point.
cot 1
2
1
2
sec 5
1 5
csc 5
2
5
2
tan 2
1 2
cos 1
5
5
5
sin 2
5
2 5
5
x 1, y 2, r 5
1, 2 y 2 x that lies in
To find a point on the terminal side of , 28. Let
Quadrant IV
tan 3
4tan
y
x
43 x
x
4
3
sec 5
3cos
x
r
x
53 x
3
5
csc 5
4sin
y
r
43 x
53 x
4
5
r
x
2 16
9 x 2
5
3 x
x , 4
3 x ,
4 x 3 y 0⇒ y 4
3 x
x > 0.
29.
sin y
r 0
x , y 1, 0, r 1 30.
since corresponds to 0, 1.3
2
csc3
2
r
y
1
1 1 31.
sec undefined3
2
r
x
1
0 ⇒
x , y 0, 1, r 1
32.
since corresponds to 1, 0.3
2
sec r
x
1
1 1 33.
sin
2
y
r 1
x , y 0, 1, r 1 34. (undefined)
since corresponds to 1, 0.
cot x
y1
0
35.
undefinedcsc r
y
1
0 ⇒
x , y 1, 0, r 1 36.
since corresponds to 0, 1.
2
cot
2 x
y 0
1 0
37.
′θ
203°
x
y
203 180 23
203 38.
′θ
309°
x
y
360 309 51
309
8/10/2019 Chapter 4 Trig Other Book
http://slidepdf.com/reader/full/chapter-4-trig-other-book 33/103
45. Quadrant III
tan 225 tan 45 1
cos 225 cos 45 2
2
sin 225 sin 45 2
2
225, 360 225 45,
47. is coterminal with
Quadrant I
tan 750 tan 30 3
3
cos 750 cos 30 3
2
sin 750 sin 30 1
2
30,
30. 750
46. Quadrant IV
sin
cos
tan 300 tan 60 3
300 cos 60 1
2
300 sin 60 3
2
300, 360 300 60,
48. is coterminal with
Quadrant IV
sin
cos
tan405 tan 45 1
405 cos 45 2
2
405 sin 45 2
2
360
315
45,
315. 405
366 Chapter 4 Trigonometry
39.
′θ
−245°
x
y
180 115 65
360 245 115 coterminal angle
245 40. is coterminal with
′θ
−145°
x
y
215 180 35
215. 145
41.
2
3
3
′θ
2
3
π
x
y
2
342.
2 7
4
4
′θ
7
4
π
x
y 7
4
43.
3.5
′θ
3.5
x
y 3.5 44. is coterminal
with
2 5
3
3
5
3.
′θ
x
y
11
3
π
11
3
8/10/2019 Chapter 4 Trig Other Book
http://slidepdf.com/reader/full/chapter-4-trig-other-book 34/103
Section 4.4 Trigonometric Functions of Any Angle 367
49. is coterminal with
Quadrant III
tan150 tan 30 3
3
cos150 cos 30 3
2
sin150 sin 30 1
2
210 180 30,
210. 150 50. .
sin
cos
tan840 tan 60 3
840 cos 60 1
2
840 sin 60 3
2
240 180 60, Quadrant III
840 is coterminal with 240
51. Quadrant III
tan4
3 tan
3 3
cos4
3 cos
3
1
2
sin4
3 sin
3
3
2
3,
4
3, 52. Quadrant I
sin
cos
tan
4 1
4 2
2
4 2
2
4,
4, 53. Quadrant IV
tan
6 tan
6
3
3
cos
6 cos
6 3
2
sin
6 sin
6
1
2
6,
6,
54. is coterminal with
sin
cos
tan is undefined.
2 tan3
2
2 cos3
2 0
2 sin3
2 1
3
2.
255. is coterminal with
Quadrant II
tan11
4 tan
4 1
cos11
4 cos
4
2
2
sin11
4 sin
4 2
2
3
4
4,
3
4 .
11
4 56. is coterminal with
Quadrant III
sin
cos
tan10
3 tan
3 3
10
3 cos
3
1
2
10
3 sin
3
3
2
4
3
3,
4
3.
10
3
57. is coterminal with
tan3
2 tan
2 which is undefined.
cos3
2 cos
2 0
sin3
2 sin
2 1
2.
2,
3
2 58. .
tan
25
4 tan
4 1
cos25
4 cos
4 2
2
sin25
4 sin
4 2
2
2 7
4
4 in Quadrant IV.
25
4 is coterminal with
7
4
8/10/2019 Chapter 4 Trig Other Book
http://slidepdf.com/reader/full/chapter-4-trig-other-book 35/103
74. cot 1.35 1
tan 1.35 0.2245
77. sin0.65 0.6052 78. sec 0.29 1
cos 0.29 1.0436
79. cot11
8 1
tan11
8 0.4142 80. csc15
14 1
sin15
14 4.4940
368 Chapter 4 Trigonometry
59.
in Quadrant IV.
cos 4
5
cos > 0
cos2 16
25
cos2 1 9
25
cos2 1 3
52
cos2 1 sin2
sin2 cos2 1
sin 3
560. cot
sin 1
csc
1
10 10
10
csc 1
sin
10 csc
csc > 0 in Quadrant II.
10 csc2
1 32 csc2
1 cot2 csc2
3 61.
in Quadrant III.
sec 13
2
sec < 0
sec2 13
4
sec2
1
9
4
sec2 1 3
22
sec2 1 tan2
tan 3
2
62.
cot 3
cot < 0 in Quadrant IV.
cot2 3
cot2 22 1
cot2 csc2 1
1 cot2 csc2
csc 2 63.
sec 1
58
8
5
cos
1
sec ⇒ sec
1
cos
cos 5
8 64.
.
tan 65
4
tan > 0 in Quadrant III
tan2 65
16
tan2 9
42
1
tan2 sec2 1
1 tan2 sec2
sec 9
4
65. sin 10 0.1736 66. sec 225
1
cos 225 1.4142 67. cos110 0.3420
68. csc330 1
sin330 2.0000 69. tan 304 1.4826 70. cot 178
1
tan 178 28.6363
71. sec 72 1
cos 72 3.2361 72. tan188 0.1405 73. tan 4.5 4.6373
75. tan
9 0.3640 76. tan
9 0.3640
8/10/2019 Chapter 4 Trig Other Book
http://slidepdf.com/reader/full/chapter-4-trig-other-book 36/103
Section 4.4 Trigonometric Functions of Any Angle 369
81. (a) reference angle is 30 or is in Quadrant I or Quadrant II.
Values in degrees: 30 , 150
Values in radians:
(b) reference angle is 30 or is in Quadrant III or Quadrant IV.
Values in degrees: 210 , 330
Values in radians:7
6,
11
6
6and sin
1
2 ⇒
6,
5
6
6and sin
1
2 ⇒
82. (a) cos reference angle is 45 or and is in Quadrant I or IV.
Values in degrees: 45 , 315
Values in radians:
(b) cos reference angle is 45 or and is in Quadrant II or III.
Values in degrees: 135 , 225
Values in radians:3
4,
5
4
4
2
2 ⇒
4,
7
4
4
2
2 ⇒
83. (a) reference angle is 60 or and is in Quadrant I or Quadrant II.
Values in degrees: 60 , 120
Values in radians:
(b) reference angle is 45 or and is in Quadrant II or Quadrant IV.
Values in degrees: 135 , 315
Values in radians:3
4,
7
4
4
cot 1 ⇒
3,
2
3
3csc
2 3
3 ⇒
84. (a) sec reference angle is 60 or and is in
Quadrant I or IV.
Values in degrees: 60 , 300
Values in radians:
(b) sec reference angle is 60 or and is
in Quadrant II or III.
Values in degrees: 120 , 240
Values in radians:2
3,
4
3
3 2 ⇒
3
,5
3
3 2 ⇒ 85. (a) reference angle is 45 or and is in
Quadrant I or Quadrant III.
Values in degrees: 45 , 225
Values in radians:
(b) reference angle is 30 or and
is in Quadrant II or Quadrant IV.
Values in degrees: 150 , 330
Values in radians:5
6,
11
6
6cot 3 ⇒
4,
5
4
4tan 1 ⇒
8/10/2019 Chapter 4 Trig Other Book
http://slidepdf.com/reader/full/chapter-4-trig-other-book 37/103
370 Chapter 4 Trigonometry
86. (a) sin reference angle is 60 or and is
in Quadrant I or II.
Values in degrees: 60 , 120
Values in radians:
(b) sin reference angle is 60 or and
is in Quadrant III or IV.
Values in degrees: 240 , 300
Values in radians:4
3,
5
3
3
32
⇒
3,
2
3
3
3
2 ⇒ 87. (a) New York City:
Fairbanks:
(b)
(c) The periods are about the same for both models,
approximately 12 months.
F 36.641 sin0.502t 1.831 25.610
N 22.099 sin0.522t 2.219 55.008
88.
(a) For February 2006,
(b) For February 2007,
(c) For June 2006,
(d) For June 2007,
S 23.1 0.44218 4.3 cos 18
6 26,756 units
t 18.
S 23.1 0.4426 4.3 cos 6
6 21,452 units
t 6.
S 23.1 0.44214 4.3 cos 14
6 31,438 units
t 14.
S 23.1 0.4422 4.3 cos 2
6 26,134 units
t 2.
S 23.1 0.442t 4.3 cos t
689.
(a)
(b)
(c) y1
2 2 cos 3 1.98 centimeters
y1
4 2 cos3
2 0.14 centimeter
y0 2 cos 0 2 centimeters
yt 2 cos 6t
Month New York City Fairbanks
February
March
May
June
August
September
November 6.546.8
41.768.6
55.675.5
59.572.5
48.663.4
13.941.6
1.434.6
90.
(a)
centimeters
(b)
centimeters
(c)
centimeters y12 2e12 cos6 1
2 1.2
t 12
y14 2e14 cos6 1
4 0.11
t 14
y0 2e0 cos 0 2
t 0
yt 2et cos 6t 91.
ampere I 0.7 5e1.4 sin 0.7 0.79
I 5e2t sin t
8/10/2019 Chapter 4 Trig Other Book
http://slidepdf.com/reader/full/chapter-4-trig-other-book 38/103
Section 4.4 Trigonometric Functions of Any Angle 371
92. sin
(a)
miles
(b)
miles
(c)
milesd 6
sin 120 6.9
120
d 6
sin 90º
6
1 6
90
d 6
sin 30
6
12 12
30
6
d ⇒ d
6
sin 93. False. In each of the four quadrants, the sign of the secant
function and the cosine function will be the same since
they are reciprocals of each other.
94. False. For example, if and but is not the reference angle.
The reference angle would be For in Quadrant II, For in Quadrant III,
For in Quadrant IV, 360 .
180. 180 . 45.
360n 1350 ≤ 135 ≤ 360, 225,n 1
95. As increases from to x decreases from 12 cm to 0 cm and y increases from 0 cm to 12 cm.
Therefore, increases from 0 to 1 and decreases from 1 to 0. Thus,
increases without bound, and when the tangent is undefined. 90tan y
x
cos x
12sin
y
12
90,0
96. Determine the trigonometric function of the reference angle and prefix the appropriate sign.
97.
x -intercepts:
y-intercept:
No asymptotes
Domain: All real numbers x
0, 4
−2−6−8 2 4 6 8−2
−4
−8
2
4
6
8
x (1, 0)(−4, 0)
(0, −4)
4, 0, 1, 0
y x 2 3 x 4 x 4 x 1 98.
x -intercepts:
y-intercepts:
No asymptotes
Domain: All real numbers x
0, 0
−1−2−3 1 2 3 4 5−1
−2
−3
−4
1
2
y
x (0, 0) , 05
2( (
0, 0, 52, 0
y 2 x 2 5 x x 2 x 5
99.
x -intercept:
y-intercept:
No asymptotes
Domain: All real numbers x
0, 8
2, 0
−6 −4−8 2 4 6 8
−4
10
12
x
(0, 8)
(−2, 0)
f x x 3 8 100.
x -intercepts:
y-intercepts:
No asymptotes
Domain: All real numbers x
0, 3
−2−3−4 2 3 4
−3
−4
1
2
3
4
x (−1, 0)
(0, −3)
(1, 0)
1, 0, 1, 0
x 2 3 x 1 x 1
g x x 4 2 x 2 3 x 2 3 x 2 1
8/10/2019 Chapter 4 Trig Other Book
http://slidepdf.com/reader/full/chapter-4-trig-other-book 39/103
8/10/2019 Chapter 4 Trig Other Book
http://slidepdf.com/reader/full/chapter-4-trig-other-book 40/103
Section 4.5 Graphs of Sine and Cosine Functions 373
Section 4.5 Graphs of Sine and Cosine Functions
■ You should be able to graph
■ Amplitude:
■Period:
■ Shift: Solve
■ Key increments: (period)1
4
bx c 0 and bx c 2 .
2
b
a
y a sinbx c and y a cosbx c. Assume b > 0.
Vocabulary Check
1. cycle 2. amplitude
3. 4. phase shift
5. vertical shift
2
b
105.
Domain: All real numbers except
x -intercepts:
Vertical asymptote: x 0
±1, 0
x 0
−12 −9 −6 −3 3 6 9 12
6
9
12
(−1, 0) (1, 0) x
y y ln x 4
106.
To find the x -intercept, let
x -intercepts:
To find the y-intercept, let
y-intercepts:
The vertical asymptote is the horizontal asymptote of translated two units to the left.
Vertical asymptote:
Domain: All real numbers such that x > 2 x
x 2
y log10 x
0, 0.301
y log10
x 2 log10
2 0.301
x 0:
1, 0
0 log10
x 2 ⇒ 10 x 2 ⇒ x 1
y 0:3
2
−1
−2
−3
321−1−3
(−1, 0) x
y y log10
x 2
8/10/2019 Chapter 4 Trig Other Book
http://slidepdf.com/reader/full/chapter-4-trig-other-book 41/103
374 Chapter 4 Trigonometry
7.
Period:
Amplitude: 2 2
2
1 2
y 2 sin x 8.
Period:
Amplitude: a 1 1
2
b
2
23 3
y cos2 x
3 9.
Period:
Amplitude: 3 3
2
10
5
y 3 sin 10 x
10.
Period:
Amplitude: a 1
3
2
b
2
8
4
y 1
3 sin 8 x 11.
Period:
Amplitude: 1
2 1
2
2
23 3
y 1
2 cos
2 x
3 12.
Period:
Amplitude: a 5
2
2
b
2
14 8
y 5
2 cos
x
4
13.
Amplitude: 1
4 1
4
Period:2
2 1
y 1
4 sin 2 x 14.
Period:
Amplitude: a 2
3
2
b
2
10 20
y 2
3 cos
x
10 15.
The graph of g is a horizontal shift
to the right units of the graph of
f a phase shift.
g x sin x
f x sin x
16.
g is a horizontal shift of f units
to the left.
f x cos x , g x cos x 17.
The graph of g is a reflection in
the x -axis of the graph of f.
g x cos 2 x
f x cos 2 x 18.
g is a reflection of f about the
y-axis.
f x sin 3 x , g x sin3 x
19.
The period of f is twice that of g.
g x cos 2 x
f x cos x 20.
The period of g is one-third the
period of f .
f x sin x , g x sin 3 x 21.
The graph of g is a vertical shift
three units upward of the graph of f.
f x 3 sin 2 x
f x sin 2 x
1.
Period:
Amplitude: 3 3
2
2
y 3 sin 2 x 2.
Period:
Amplitude: a 2
2
b
2
3
y 2 cos 3 x 3.
Period:
Amplitude: 5
2 5
2
2
12 4
y 5
2 cos
x
2
4.
Period:
Amplitude: a 3 3
2
b
2
13 6
y 3 sin x
35.
Period:
Amplitude: 1
2 1
2
2
3 6
y 1
2 sin
x
3 6.
Period:
Amplitude: a 3
2
2
b
2
2 4
y 3
2 cos
x
2
8/10/2019 Chapter 4 Trig Other Book
http://slidepdf.com/reader/full/chapter-4-trig-other-book 42/103
Section 4.5 Graphs of Sine and Cosine Functions 375
28.
Period:
Amplitude: 1
Symmetry: origin
Key points: Intercept Maximum Intercept Minimum Intercept
Since the graph of is the graph of but stretched horizontally by a factor of 3.
Generate key points for the graph of by multiplying the coordinate of each key point of by 3. f x x -g x
f x ,g x g x sin x 3 f x
3,
2 , 03
2, 1 , 0 2
, 10, 0
2
b
2
1 2
− 2
2
π 6
f g
x
y f x sin x
29.
Period:
Amplitude: 1
Symmetry: axis
Key points: Maximum Intercept Minimum Intercept Maximum
Since the graph of is the graph of but translated upward by one unit.
Generate key points for the graph of by adding 1 to the coordinate of each key point of f x . y-g x f x ,g x g x 1 cos x f x 1,
2 , 13
2 , 0 , 1
2, 00, 1
y-
2
b
2
1 2
−1
g
f
x π π 2
y f x cos x
25. The graph of g is a horizontal shift units to the right of
the graph of f.
26. Shift the graph of f two units upward to obtain the graph
of g.
27.
Period:
Amplitude: 2
Symmetry: origin
Key points: Intercept Minimum Intercept Maximum Intercept
Since generate key points for the graph of by multiplying
the coordinate of each key point of by
2. f x y-
g x g x 4 sin x 2 f x ,
2 , 03
2, 0 , 0 2
, 20, 0
2
b
2
1 2
x
−
π π 32 2
5
4
3
−5
f
g
y f x 2 sin x
22.
g is a vertical shift of f two units
downward.
f x cos 4 x , g x 2 cos 4 x 23. The graph of g has twice the
amplitude as the graph of f. The
period is the same.
24. The period of g is one-third the
period of f .
8/10/2019 Chapter 4 Trig Other Book
http://slidepdf.com/reader/full/chapter-4-trig-other-book 43/103
376 Chapter 4 Trigonometry
31.
Period:
Amplitude:
Symmetry: origin
Key points: Intercept Minimum Intercept Maximum Intercept
Since the graph of is the graph of but translated upward by three units.
Generate key points for the graph of by adding 3 to the coordinate of each key point of f x . y-g x
f x ,g x g x 3 1
2 sin
x
2 3 f x ,
4 , 03 ,1
22 , 0 , 1
20, 0
1
2
2
b
2
12 4
−1
1
2
3
4
5
f
g
x −π π 3
y f x 1
2 sin
x
2
32.
Period:
Amplitude: 4
Symmetry: origin
Key points: Intercept Maximum Intercept Minimum Intercept
Since the graph of is the graph of but translated downward by three units.
Generate key points for the graph of by subtracting 3 from the coordinate of each key point of f x . y-g x f x ,g x g x 4 sin x 3 f x 3,
2, 03
2, 21, 01
2, 20, 0
2
b
2
2
−8
2
4 f
g
1 x
y f x 4 sin x
30.
Period:
Amplitude: 2
Symmetry: axis
Key points: Maximum Intercept Minimum Intercept Maximum
Since the graph of is the graph of but
i) shrunk horizontally by a factor of 2,
ii) shrunk vertically by a factor of and
iii) reflected about the axis.
Generate key points for the graph of by
i) dividing the coordinate of each key point of by 2, and
ii) dividing the coordinate of each key point of by 2. f x y-
f x x -
g x
x -
12,
f x ,g x g x cos 4 x 12 f 2 x ,
, 23
4, 0 2
, 2 4, 00, 2
y-
2
b
2
2
−2
2
π
f
g
x
y f x 2 cos 2 x
8/10/2019 Chapter 4 Trig Other Book
http://slidepdf.com/reader/full/chapter-4-trig-other-book 44/103
Section 4.5 Graphs of Sine and Cosine Functions 377
35.
Period:
Amplitude:
Key points:
3
2, 3, 2 , 0
0, 0, 2, 3, , 0,
3
2
y
x π π 32 2
−−
π
2π 32
−4
1
2
3
4
y 3 sin x
37.
Period:
Amplitude:
Key points:
3
2, 0, 2 ,
1
3
0,1
3, 2, 0, , 1
3,
13
2
y
x π π
2π 2
1
−1
2
3
4
3
1
32
3
4
3
−
−
−
y 1
3 cos x 38.
Period:
Amplitude:
Key points:
3
2, 0, 2 , 4
0, 4, 2, 0, , 4,
4
2
y
x π π 2π −2 −π
−2
−4
4
y 4 cos x
36.
Period:
Amplitude:
Key points:
3
2,
1
4, 2 , 0
0, 0, 2,
1
4, , 0,
1
4
2
y
x π π 2π −2 −π
−1
−2
1
2
y 1
4 sin x
33.
Period:
Amplitude: 2
Symmetry: axis
Key points: Maximum Intercept Minimum Intercept Maximum
Since the graph of is the graph of but with a phase shift (horizontal translation)
of Generate key points for the graph of by shifting each key point of units to the left. f x g x .
f x ,g x g x 2 cos x f x ,
2 , 23
2, 0 , 2 2
, 00, 2
y-
2
b
2
1 2
−3
3
f
g
x
π π 2
y f x 2 cos x
34.
Period:
Amplitude: 1
Symmetry: axis
Key points: Minimum Intercept Maximum Intercept Minimum
Since the graph of is the graph of but with a phase shift (horizontal translation)
of Generate key points for the graph of by shifting each key point of units to the right. f x g x .
f x ,g x g x cos x f x ,
2 , 13
2, 0 , 1 2
, 00, 1
y-
2
b
2
1 2
−2
2
π π 2
f
g
x
y f x cos x
8/10/2019 Chapter 4 Trig Other Book
http://slidepdf.com/reader/full/chapter-4-trig-other-book 45/103
378 Chapter 4 Trigonometry
45.
Period:
Amplitude: 1
Shift: Set
Key points: 4, 0, 3
4, 1, 5
4, 0, 7
4, 1, 9
4, 0
x
4 x
9
4
x
4 0 and x
4 2
2
−3
−2
1
2
3
π π −
x
y y sin x
4; a 1, b 1, c
4
44.
Period:
Amplitude: 10
Key points:
1284−4−12
12
8
4
−12
x
y
0, 10, 3, 0, 6, 10, 9, 0, 12, 10
2
6 12
y 10 cos x
643.
Period:
Amplitude: 1
Key points:
−1 2 3
−3
−2
2
3
x
y
0, 0, 3
4, 1, 3
2, 0, 9
4, 1, 3, 0
2
2 3 3
y sin2 x
3; a 1, b
2
3, c 0
39.
Period:
Amplitude: 1
Key points:
3 , 0, 4 , 10, 1,
, 0, 2
,
1,
4
y
x π 4π 2π −2
−1
−2
2
y cos x
2 40.
Period:
Amplitude: 1
Key points:
3
8, 1, 2
, 0
0, 0,
8, 1,
4, 0,
2
y
x
−2
1
2
π
4
y sin 4 x
41.
Period:
Amplitude: 1
Key points:
0, 1, 1
4, 0, 1
2, 1, 3
4, 0
2
2 1
1 2
−2
1
2
x
y y cos 2 x 42.
Period:
Amplitude: 1
Key points:
6, 1, 8, 0
0, 0, 2, 1, 4, 0,
2
4 8
2
1
−2
62−2−6 x
y y sin x
4
8/10/2019 Chapter 4 Trig Other Book
http://slidepdf.com/reader/full/chapter-4-trig-other-book 46/103
8/10/2019 Chapter 4 Trig Other Book
http://slidepdf.com/reader/full/chapter-4-trig-other-book 47/103
380 Chapter 4 Trigonometry
55.
Period:
Amplitude:
Shift:
Key points: 2,
2
3, 3
2, 0, 5
2, 2
3 , 7
2, 0, 9
2,
2
3
x
2 x
9
2
x
2
4 0 and
x
2
4 2
2
3
4
−4
−3
−2
−1
1
2
3
4
π π 4 x
y
y
2
3 cos x
2
4; a
2
3, b
1
2, c
4
53.
Period:
Amplitude: 3
Shift: Set
Key points: , 0,
2, 3, 0, 6, 2
, 3, , 0
x x
x 0 and x 2
2
−8
2
4
π π 2 x
y y 3 cos x 3
54.
Period:
Amplitude: 4
Shift: Set
Key points:
4, 8, 4
, 4, 3
4, 0, 5
4, 4, 7
4, 8
x
4 x
7
4
x
4 0 and x
4 2
2
−4
2
4
6
10
x
y
π 3π 2π −2 − π π
y
4 cos x
4
4
51.
Period:
Amplitude:
Vertical shift two units upwardKey points:
0.20.1−0.1 0
1.8
2.2
x
y
0, 2.1, 1
120, 2, 1
60, 1.9, 1
40, 2, 1
30, 2.1
1
10
2
60
1
30
y 2 1
10 cos 60 x 52.
Period:
Amplitude: 2
Key points:
−7
−6
−5
−4
1
−
x
y
π π π 2
0, 1,
2, 3, , 5, 3
2 , 3, 2 , 1
2
y 2 cos x 3
8/10/2019 Chapter 4 Trig Other Book
http://slidepdf.com/reader/full/chapter-4-trig-other-book 48/103
Section 4.5 Graphs of Sine and Cosine Functions 381
63.
Amplitude:
Vertical shift one unit upward of
Thus, f x 2 cos x 1.
g x 2 cos x ⇒ d 1
1
23 1 2 ⇒ a 2
f x a cos x d 64.
Amplitude:
a 2, d 1
d 1 2 1
1 2 cos 0 d
1 32
2
f x a cos x d
65.
Amplitude:
Since is the graph of reflected in the
x -axis and shifted vertically four units upward, we haveThus, f x 4 cos x 4.a 4 and d 4.
g x 4 cos x f x
1
28 0 4
f x a cos x d 66.
Amplitude:
Reflected in the axis:
a 1, d 3
d 3
4 1 cos 0 d
a 1 x -
2 4
2 1
f x a cos x d
56.
Period:
Amplitude: 3
Shift: Set
Key points:
6, 3,
12, 0, 0, 3, 12
, 0, 6, 3
x
6 x
6
6 x 0 and 6 x 2
2
6
3
2
3
x
y
π
y 3 cos6 x
57.
−6 6
−4
4
y 2 sin4 x 58.
−12
−8
12
8
y 4 sin2
3 x
3 59.
−3
−1
3
3
y cos2 x
2 1
60.
−6
−6
6
2
y 3 cos x
2
2 2 61.
−20
−0.12
20
0.12
y 0.1 sin x
10 62.
−0.03
−0.02
0.03
0.02
y 1
100 sin 120 t
8/10/2019 Chapter 4 Trig Other Book
http://slidepdf.com/reader/full/chapter-4-trig-other-book 49/103
382 Chapter 4 Trigonometry
74.
(a) Period
(b)1 cycle
4 seconds
60 seconds
1 minute 15 cycles per minute
2
2 4 seconds
v
1.75 sin
t
2
71.
In the interval
when x 5
6,
6,
7
6,
11
6.sin x
1
2
2 , 2 ,
y2
1
2
−2
2
2
−2
y1 sin x 72.
y1 y
2 when x ,
y2 1
−2
2
2
−2
y1 cos x
73.
(a) Time for one cycle
(b) Cycles per min cycles per min
(c) Amplitude: 0.85; Period: 6
Key points: 0, 0, 3
2, 0.85, 3, 0, 9
2, 0.85, 6, 0
60
6 10
2
3 6 sec
t 2 4 8 10
0.25
0.50
0.75
1.00
−0.25
−1.00
v y 0.85 sin t
3
(c)
1 3 5 7
−2
−3
1
2
3
t
v
69.
Amplitude:
Period:
Phase shift:
Thus, y 2 sin x
4.
1
4 c 0 ⇒ c
4
bx c 0 when x
4
2 ⇒ b 1
a 2
y a sinbx c 70.
Amplitude:
Period: 2
Phase shift:
a 2, b
2, c
2
c
b
1 ⇒ c
2
2
b 4 ⇒ b
2
2 ⇒ a 2
y a sinbx c
67.
Amplitude:
Since the graph is reflected in the x -axis, we have
Period:
Phase shift:
Thus, y 3 sin 2 x .
c 0
2
b ⇒ b 2
a 3.
a 3
y a sinbx c 68.
Amplitude:
Period:
Phase shift:
a 2, b 1
2, c 0
c 0
2
b 4 ⇒ b
1
2
4
2 ⇒ a 2
y a sinbx c
8/10/2019 Chapter 4 Trig Other Book
http://slidepdf.com/reader/full/chapter-4-trig-other-book 50/103
Section 4.5 Graphs of Sine and Cosine Functions 383
78. (a) and (c)
Reasonably good fit
(d) Period is 29.6 days.
(e) March 12
The Naval observatory says that 50% of the moon’s face will be illuminated on
March 12, 2007.
y 0.44 44% ⇒ x 71.
y
x 10 20 30 40
0.2
0.4
0.6
0.8
1.0
P e r c e n t o f m o o n ’ s
f a c e i l l u m i n a t e d
Day of the year
(b)
y 12 1
2 sin0.21 x 0.92
C 0.92
Horizontal shift: 0.213 7.4 C 0
b 2
29.6 0.21
2
b 47.4 29.6
Period: 8 8 7 6 8
5 7.4 (average length of interval in data)
Amplitude: 12
⇒ a 12
Vertical shift: 1
2 ⇒ d
1
2
75.
(a) Period:
(b) f 1
p 440 cycles per second
2
880
1
440 seconds
y 0.001 sin 880 t 76.
(a) Period:
(b)1 heartbeat
65 seconds
60 seconds
1 minute 50 heartbeats per minute
2
5 3
6
5 seconds
P 100 20 cos5 t
3
77. (a)
(b)
The model is a good fit.
0
0
12
100
C t 56.55 26.95 cos t
6 3.67
d 1
2high low
1
283.5 29.6 56.55
c
b 7 ⇒ c 7 6 3.67
b 2
p
2
12
6
p 2high time low time 27 1 12
a 1
2high low
1
283.5 29.6 26.95 (c)
The model is a good fit.
(d) Tallahassee average maximum:
Chicago average maximum:
The constant term, gives the average maximum
temperature.
(e) The period for both models is months.
This is as we expected since one full period is one
year.
(f) Chicago has the greater variability in temperature
throughout the year. The amplitude, a, determines this
variability since it is .12high temp low temp
2
6 12
d ,
56.55
77.90
0
0
12
100
8/10/2019 Chapter 4 Trig Other Book
http://slidepdf.com/reader/full/chapter-4-trig-other-book 51/103
384 Chapter 4 Trigonometry
85. Since the graphs are the
same, the conjecture is that
.sin x cos x
22
1
−2
f = g
x π π 32 2
−
π 32
y
86. f x sin x , g x cos x
2
x 0
0 1 0 0
0 1 0 01cos x
2
1sin x
2 3
2
2
83. True.
Since and so is a reflection in the x -axis of y sin x
2.cos x sin x
2, y cos x sin x
2,
84. Answers will vary.
Conjecture: sin x cos x
22
1
−2
f = g
x
y
π π 32 2
π 32
−
79.
(a) Period
Yes, this is what is expected because there are
365 days in a year.(b) The average daily fuel consumption is given by the
amount of the vertical shift (from 0) which is given
by the constant 30.3.
(c)
The consumption exceeds 40 gallons per day when
124 < x < 252.
0
0
365
60
2
2
365
365
C 30.3 21.6 sin2 t
365 10.9 80. (a) Period
The wheel takes 12 minutes to revolve once.
(b) Amplitude: 50 feet
The radius of the wheel is 50 feet.
(c)
0
0 20
110
2
6 12 minutes
81. False. The graph of is the graph oftranslated to the left by one period, and the graphs are
indeed identical.
sin( x )sin( x 2 ) 82. False. has an amplitude that is half that
of For y a cos bx , the amplitude is a. y cos x .
y 12 cos 2 x
8/10/2019 Chapter 4 Trig Other Book
http://slidepdf.com/reader/full/chapter-4-trig-other-book 52/103
Section 4.5 Graphs of Sine and Cosine Functions 385
87. (a)
The graphs are nearly the same for
(b)
The graphs are nearly the same for
2 < x <
2.
−2
−2 2
2
2 < x <
2.
−2
−2 2
2 (c)
The graphs now agree over a wider range, 3
4 < x <
3
4.
−2
−2 2
2
−2
2 −2
2
cos x 1 x 2
2!
x 4
4!
x 6
6!
sin x x x 3
3!
x 5
5!
x 7
7!
88. (a)
(by calculator)
(c)
(by calculator)
(e)
(by calculator)cos 1 0.5403
cos 1 1 1
2!
1
4! 0.5417
sin
6 0.5
sin
6 1
63
3!
65
5! 0.5000
sin1
2 0.4794
sin1
2
1
2
123
3!
125
5! 0.4794 (b)
(by calculator)
(d)
(by calculator)
(f)
(by calculator)cos
4 0.7071
cos
4 1
42
2!
42
4! 0.7074
cos0.5 0.8776
cos0.5 1 0.52
2!
0.54
4! 0.8776
sin 1 0.8415
sin 1 1 1
3!
1
5! 0.8417
The error in the approximation is not the same in each case. The error appears to increase as x moves farther away from 0.
89. log10
x 2 log10
x 212 12 log
10 x 2
91. lnt 3
t 1 ln t 3 lnt 1 3 ln t lnt 1
93.
log10
xy
12 log
10 x log
10 y
12 log
10 xy
90.
2 log2 x log
2 x 3
log2 x 2 x 3 log
2 x 2 log
2 x 3
92.
1
2 ln z
1
2 ln z2 1
ln z
z2 1
1
2 ln z
z2 1 1
2ln z ln z2 1
94.
log2 x 3 y
log2 x 2( xy)
2 log2 x log
2 xy log
2 x 2 log
2 xy
95.
ln3 x
y4ln 3 x 4 ln y ln 3 x ln y4
8/10/2019 Chapter 4 Trig Other Book
http://slidepdf.com/reader/full/chapter-4-trig-other-book 53/103
386 Chapter 4 Trigonometry
96.
ln x 2 2 x
ln x 3 2 x
x 2
ln 2 x
x 2 ln x 3
1
2ln2 x
x 2 ln x 3
1
2ln 2 x 2 ln x 3 ln x
1
2ln 2 x ln x 2 ln x 3
Section 4.6 Graphs of Other Trigonometric Functions
■ You should be able to graph
■ When graphing or you should first graph orbecause
(a) The x -intercepts of sine and cosine are the vertical asymptotes of cosecant and secant.
(b) The maximums of sine and cosine are the local minimums of cosecant and secant.
(c) The minimums of sine and cosine are the local maximums of cosecant and secant.
■ You should be able to graph using a damping factor.
y a sinbx c y a cosbx c y a cscbx c y a secbx c
y a cscbx c y a secbx c
y a cotbx c y a tanbx c
97. Answers will vary.
1.
Period:
Matches graph (e).
2
2
y sec 2 x 3.
Period:
Matches graph (a).
1
y 1
2 cot x 2.
Period:
Asymptotes:
Matches graph (c).
x , x
b
12 2
y tan x
2
4.
Period:
Matches graph (d).
2
y csc x 5.
Period:
Asymptotes:
Matches graph (f).
x 1, x 1
2
b
2
2 4
y 12
sec
x 2
6.
Period:
Asymptotes:
Reflected in x -axis
Matches graph (b).
x 1, x 1
2
b
2
2 4
y 2 sec
x 2
Vocabulary Check
1. vertical 2. reciprocal 3. damping 4.
5. 6. 7. 2
,
1
1,
x
n
8/10/2019 Chapter 4 Trig Other Book
http://slidepdf.com/reader/full/chapter-4-trig-other-book 54/103
Section 4.6 Graphs of Other Trigonometric Functions 387
7.
Period:
Two consecutive asymptotes:
x
2 and x
2
1
2
3
x π π
y
−
y 1
3 tan x
9.
Period:
Two consecutive asymptotes:
3 x
2 ⇒ x
6
3 x
2 ⇒ x
6
3
x
4
3
2
1
− π
3π
3
y y tan 3 x
x 0
y 01
3
1
3
4
4
x 0
y 0 11
12
12
8.
Period:
Two consecutive asymptotes:
x
2, x
2
−3
1
2
3
π π −
x
y y 1
4 tan x
x 0
y 01
4
1
4
4
4
10.
Period:
Two consecutive
asymptotes:
x 12
, x 12
1
2
−8
−4
x
y
y 3 tan x
x 0
y 3 0 3
1
4
1
4
13.
Period:
Two consecutive
asymptotes:
x 0, x 1
2
2
x
4
3
2
1
−3
−4
21−1−2
y y csc x
11.
Period:
Two consecutiveasymptotes:
x
2, x
2
2
1
2
3
x π
y
− π
y 1
2 sec x 12.
Period:
Two consecutive
asymptotes:
x
2, x
2
2
π
3
2 x
y
y 1
4 sec x
14.
Period:
Two consecutive
asymptotes:
x 0, x
4
2
4
2
8
6
4
2
−2− π
4π
4
x
y y 3 csc 4 x
0
11
21 y
3
3 x 0
1
2
1
4
1
2 y
3
3 x
2 1 2 y
5
6
1
2
1
6 x
6 3 6 y
5
24
8
24 x
8/10/2019 Chapter 4 Trig Other Book
http://slidepdf.com/reader/full/chapter-4-trig-other-book 55/103
8/10/2019 Chapter 4 Trig Other Book
http://slidepdf.com/reader/full/chapter-4-trig-other-book 56/103
8/10/2019 Chapter 4 Trig Other Book
http://slidepdf.com/reader/full/chapter-4-trig-other-book 57/103
390 Chapter 4 Trigonometry
39.
−6
−0.6
6
0.6 y 0.1 tan x
4
4
41.
x π π 2
2
y
x 7
4,
3
4,
4,
5
4
tan x 1
40.
−6
−2
6
2
y 1
3 cos x
2
2 ⇒ y
1
3 sec x
2
2
42.
x π π 2
2
1
x 5
3,
2
3,
3,
4
3
tan x 3 43.
x
y
π
2
3π
2
1
2
3
−3
x 4
3,
3,
2
3,
5
3
cot x 3
3
31.
−5
5 −5
5
y tan x
3 33.
−4
2
2−
4
y 2 sec 4 x 2
cos 4 x 32.
−3
3
4
3
4
3−
y tan 2 x
34.
−3
−2
3
2
y sec x ⇒ y 1
cos x 35.
−3
−
2
3
2
3
3
y tan x
4 36.
−3
2
3
2
3−
3
1
4 tan x
2
y 1
4 cot x
2
37.
y 1
sin4 x
−3
−
2
2
3 y csc4 x 38.
y 2
cos2 x
−4
−
4 y 2 sec(2 x ) ⇒
8/10/2019 Chapter 4 Trig Other Book
http://slidepdf.com/reader/full/chapter-4-trig-other-book 58/103
Section 4.6 Graphs of Other Trigonometric Functions 391
44.
x π
2
π
2
3π
23π
2− −
2
3
−3
x 7
4,
3
4,
4,
5
4
cot x 1 45.
x π π 2π π 2
1
−−
y
x ±2
3, ±
4
3
sec x 2 46.
x π π π 2π 2
1
−−
y
x 5
3,
3,
3,
5
3
sec x 2
47.
x 7
4,
5
4,
4,
3
4
x π 3ππ3π
2 2 2 2− −
1
2
3
−1
ycsc x 2 48.
x 2
3,
3,
4
3,
5
3
x π 3ππ3π
2 2 2 2− −
1
2
3
csc x 2 3
3
49.
Thus, is an even function and the graph has
axis symmetry. y-
f x sec x
f x
1
cos x
1
cos x
f x sec x
y
x
3
4
π π π 2−
f x sec x 1
cos x 50.
Thus, the function is odd
and the graph of
is symmetric about the origin.
y tan x
tan x tan x
x
2
3
−3
3π
2−
π
2−
π
2
3π
2
f x tan x
51.
(a)
(b) on the interval,
(c) As and
since is
the reciprocal of f x .g x g x
12 csc x → ±
f x 2 sin x → 0 x → ,
6< x <
5
6 f > g
1
−1
2
3
f
g
x π π π 32 4
π
4
y
g x 1
2 csc x
f x 2 sin x 52.
(a)
(b) The interval in which
(c) The interval in which
which is the same interval as part (b).
2 f < 2g is 1,13,
f < g is 1,13.
−3
g
f
3
1−1
f x tan x
2, g x
1
2 sec
x
2
8/10/2019 Chapter 4 Trig Other Book
http://slidepdf.com/reader/full/chapter-4-trig-other-book 59/103
392 Chapter 4 Trigonometry
57.
As
Matches graph (d).
x → 0, f x → 0 and f x > 0.
f x x cos x 58.
Matches graph (a) as x → 0, f x → 0.
f x x sin x
60.
Matches graph (c) as x → 0, g x → 0.
g x x cos x 59.
As
Matches graph (b).
x → 0, g x → 0 and g x is odd.
g x x sin x
61.
The graph is the line y 0.
f x g x
g x 0
−3 −2 −1 1 2 3
−3
−2
−1
1
2
3
x
y f x sin x cos x
2 62.
It appears that
That is,
sin x cos x
2 2 sin x .
f x g x .
g x 2 sin x
−4
2
4
x π π −
y
f x sin x cos x
2
63.
f x g x
g x 1
21 cos 2 x
–1
2
3
y
x π π −
f x sin2 x 64.
It appears that
That is,
cos2 x
2
1
21 cos x .
f x g x .
g x 1
21 cos x
−1
−3 3 6−6
2
3
x
y f x cos2 x
2
53.
The expressions are equivalent except when
and y1 is undefined.
sin x 0
sin x csc x sin x 1
sin x 1, sin x 0
−2
3−3
2
y1 sin x csc x and y
2 1 54.
The expressions are equivalent.
sin x sec x sin x 1
cos x
sin x
cos x tan x
−4
−2 2
4
y1 sin x sec x , y
2 tan x
55.
The expressions are equivalent.
−4
2 −2
4cot x cos x
sin x
y1
cos x
sin x and y
2 cot x
1
tan x 56.
The expressions are equivalent.
tan2 x sec2 x 1
1 tan2 x sec2 x
−1
−
3
2
3
2
3
y1 sec2 x 1, y
2 tan2 x
8/10/2019 Chapter 4 Trig Other Book
http://slidepdf.com/reader/full/chapter-4-trig-other-book 60/103
Section 4.6 Graphs of Other Trigonometric Functions 393
65.
The damping factor is
As x → , g x →0.
y e x 22.
e x 22≤ g x ≤ e x 22
−1
8−8
1g x e x 22 sin x
66.
Damping factor:
As x → , f x → 0.
−3 6
−3
3
e x
f x e x cos x 67.
Damping factor:
As x →, f x → 0.
−9 9
−6
6
y 2 x 4.
2 x 4 ≤ f x ≤ 2 x 4
f x 2 x
4
cos x 68.
Damping factor:
As x → , h x → 0.
−8
−1
8
1
2 x 24
h x 2 x 2
4 sin x
69.
As x → 0, y → .
0
−2
8
6
y 6
x cos x , x > 0 71.
As x → 0, g x → 1.
−1
6 −6
2
g x sin x
x 70.
As x → 0, y → .
0
−2
6
6
y 4
x sin 2 x , x > 0
72.
As x → 0, f ( x ) → 0.
−1
−6 6
1
f ( x ) 1 cos x
x
73.
As oscillates
between and 1.1
x → 0, f x
−2
−
2
f x sin1
x
74.
As oscillates. x → 0, h( x )
−1
−
2
h( x ) x sin1
x
75.
Grounddistance
x
14
10
6
2
−2
−6
−10
−14
Angle of elevation
d
π π π 32 4
π
4
d 7tan x
7 cot x
tan x 7
d 76.
x
20
40
60
80
Angle of camera
D i s t a n c e
d
0 π
2π
4π
4π
2− −
d 27cos x
27 sec x ,
2 < x <
2
cos x 27
d
8/10/2019 Chapter 4 Trig Other Book
http://slidepdf.com/reader/full/chapter-4-trig-other-book 61/103
8/10/2019 Chapter 4 Trig Other Book
http://slidepdf.com/reader/full/chapter-4-trig-other-book 62/103
Section 4.6 Graphs of Other Trigonometric Functions 395
83. As from the left,
As from the right, f x tan x → . x →
2
f x tan x → . x →
282. True.
If the reciprocal of is translated units to the
left, we have
y 1
sin x
2 1
cos x sec x .
2 y sin x
y sec x 1
cos x
84. As from the left, .
As from the right, . f ( x ) csc x → x →
f ( x ) csc x → x →
85.
(a)
The zero between 0 and 1 occurs at x 0.7391.
3
−2
−3
2
f x x cos x
(b)
This sequence appears to be approaching the zero
of f : x 0.7391.
x 9 cos 0.7504 0.7314
x 8 cos 0.7221 0.7504
x 7 cos 0.7640 0.7221
x 6 cos 0.7014 0.7640
x 5 cos 0.7935 0.7014
x 4 cos 0.6543 0.7935
x 3 cos 0.8576 0.6543
x 2 cos 0.5403 0.8576
x 1
cos 1 0.5403
x 0 1
x n cos x
n1
86.
The graphs are nearly the
same for 1.1 < x < 1.1.
y x 2 x 3
3!
16 x 5
5!
−6
−
2
3
2
3
6 y tan x 87.
The graph appears to
coincide on the interval
1.1 ≤ x ≤ 1.1.
y2 1
x 2
2!
5 x 4
4!
−6
−
6
2
3
2
3
y1 sec x
88. (a)
—CONTINUED—
−3 3
−2
2
y2
−3 3
−2
2
y1
y2 4
sin x 13
sin 3 x 15
sin 5 x y1 4
sin x 13
sin 3 x
8/10/2019 Chapter 4 Trig Other Book
http://slidepdf.com/reader/full/chapter-4-trig-other-book 63/103
8/10/2019 Chapter 4 Trig Other Book
http://slidepdf.com/reader/full/chapter-4-trig-other-book 64/103
Section 4.7 Inverse Trigonometric Functions 397
■ You should know the definitions, domains, and ranges of y arcsin x , y arccos x , and y arctan x .
Function Domain Range
y arcsin x ⇒ x sin y
y arccos x ⇒ x cos y
y arctan x ⇒ x tan y
■ You should know the inverse properties of the inverse trigonometric functions.
■ You should be able to use the triangle technique to convert trigonometric functions of inverse trigonometric functions
into algebraic expressions.
tanarctan x x and arctantan y y,
2 < y <
2
cosarccos x x and arccoscos y y, 0 ≤ y ≤
sinarcsin x x and arcsinsin y y,
2 ≤ y ≤
2
2 < x <
2 < x <
0 ≤ y ≤ 1 ≤ x ≤ 1
2
≤ y ≤ 2
1 ≤ x ≤ 1
Vocabulary Check
Alternative
Function Notation Domain Range
1.
2.
3.
2 < y <
2 < x < y tan1 x y arctan x
0 ≤ y ≤ 1 ≤ x ≤ 1 y cos1 x y arccos x
2 ≤ y ≤
21 ≤ x ≤ 1 y sin1 x y arcsin x
1.
2 ≤ y ≤
2 ⇒ y
6 y arcsin
1
2 ⇒ sin y
1
2 for 2. y arcsin 0 ⇒ sin y 0 for
2 ≤ y ≤
2 ⇒ y 0
3. 0 ≤ y ≤ ⇒ y
3 y arccos
1
2 ⇒ cos y
1
2 for 4. y arccos 0 ⇒ cos y 0 for 0 ≤ y ≤ ⇒ y
2
5.
2 < y <
2 ⇒ y
6
y arctan 3
3 ⇒ tan y
3
3 for 6.
2 < y <
2 ⇒ y
4
y arctan1 ⇒ tan y 1 for
Section 4.7 Inverse Trigonometric Functions
8/10/2019 Chapter 4 Trig Other Book
http://slidepdf.com/reader/full/chapter-4-trig-other-book 65/103
8/10/2019 Chapter 4 Trig Other Book
http://slidepdf.com/reader/full/chapter-4-trig-other-book 66/103
Section 4.7 Inverse Trigonometric Functions 399
36.
cos 6 3
2
arccos1
2 2
3
arccos1 37.
tan arctan x
4 θ
4
x
tan x
4
38.
arccos4
x
cos 4
x 39.
sin arcsin x 2
5 θ
5 x + 2
sin x 2
5
40.
arctan x 1
10
tan x 1
1041.
arccos x 3
2 x θ
x + 3
2 x
cos x 3
2 x
42.
x 1
arctan 1
x 1
tan x 1
x 2 1
1
x 143. sinarcsin 0.3 0.3 44. tanarctan 25 25
45. cosarccos0.1 0.1 46. sinarcsin0.2 0.2 47.
Note: 3 is not in the range of
the arcsine function.
arcsinsin 3 arcsin0 0
48.
Note: is not in the range of the arccosine function.7
2
arccoscos7
2 arccos 0
2 49. Let
and sin y 3
5.
tan y 3
4, 0 < y <
2,
x y
53
4
y
y arctan3
4,
50. Let
secarcsin4
5 sec u 5
3.
sin u 4
5, 0 < u <
2,
3
45
u
u arcsin4
5
, 51. Let
and cos y 1
5 5
5.
tan y 2 2
1, 0 < y <
2,
x
y
5 2
1
y y arctan 2,
8/10/2019 Chapter 4 Trig Other Book
http://slidepdf.com/reader/full/chapter-4-trig-other-book 67/103
400 Chapter 4 Trigonometry
52. Let
1
25
u
sinarccos 5
5 sin u 2
5
2 5
5.
cos u 5
5, 0 < u <
2,
u arccos 5
5, 53. Let
and
x y
5
12
13
y
cos y 12
13.sin y
5
13, 0 < y <
2,
y arcsin5
13,
55. Let
and
x y
34−3
5
y
sec y
345
.tan y 35
, 2 < y < 0,
y arctan3
5,54. Let
13
12
−5u
cscarctan 5
12 csc u 13
5.
tan u 512
, 2
< u < 0,
u arctan 5
12,
56. Let
4−3
7
u
tanarcsin3
4 tan u 3
7
3 7
7.
sin u 3
4,
2 < u < 0,
u arcsin3
4, 57. Let
and
x y
3
−2
5
y
sin y 5
3.cos y
2
3,
2 < y < ,
y arccos2
3,
8/10/2019 Chapter 4 Trig Other Book
http://slidepdf.com/reader/full/chapter-4-trig-other-book 68/103
Section 4.7 Inverse Trigonometric Functions 401
60. Let
x + 12
1
x
u
sinarctan x sin u x
x 2 1 .
tan u x x
1,u arctan x , 61. Let
and cos y 1 4 x 2.
sin y 2 x 2 x
1,
1 − 4 x 2
2 x
y
1
y arcsin2 x ,
62. Let
secarctan 3 x sec u 9 x 2 1.
tan u 3 x 3 x
1,
1
3 x 9 + 1 x 2
u
u arctan 3 x , 63. Let
and sin y 1 x 2.
cos y x x
1, 1 − x 2
x
y
1
y arccos x ,
64. Let
x −1
2 x x − 2
1
u
secarcsin x 1 sec u 1
2 x x 2.
sin u x 1 x 1
1,
u arcsin x 1, 65. Let
and tan y 9 x 2
x .
cos y x
3, 9 − x 2
x
y
3
y arccos x
3,
66. Let
cotarctan1
x cot u x .
tan u 1
x , 1
u x
x + 12
u arctan1
x , 67. Let
and csc y x 2 2
x .
tan y x
2,
x 2 + 2
x
y
2
y arctan x
2,
59. Let
and cot y 1
x .
tan y x x
1,
x 2 + 1 x
y
1
y arctan x ,58. Let
cotarctan5
8 cot u 8
5.
tan u 5
8, 0 < u <
2, 5
8
89
u
u arctan5
8,
8/10/2019 Chapter 4 Trig Other Book
http://slidepdf.com/reader/full/chapter-4-trig-other-book 69/103
402 Chapter 4 Trigonometry
69.
They are equal. Let
and
The graph has horizontal asymptotes at y ±1.
g x 2 x
1 4 x 2 f x
1 + 4 x 2
2 x
y
1
sin y 2 x
1 4 x 2.
tan y 2 x 2 x
1,
y arctan 2 x , −3 3
−2
2
f x sinarctan 2 x , g x 2 x
1 4 x 2
70.
Asymptote:
These are equal because:
Let
Thus, f x g x .
4 x 2
x g x
f x tanarccos x
2 tan u
u arccos x
2.
2
u x
4 − x 2
x 0
g x 4 x 2
x
−3
−2
3
2
f x tanarccos x
2 71. Let
Thus,
x 2 + 81
x
y
9
arcsin y 9
x 2 81, x < 0.
arcsin y 9
x 2 81, x > 0;
tan y 9
x and sin y
9
x 2 81, x > 0;
9
x 2 81, x < 0.
y arctan9
x .
72. If
then
arcsin 36 x 2
6 arccos x
6.
sin u 36 x 2
6,
6
u x
36 − x 2
arcsin 36 x 2
6 u, 73. Let Then,
and
Thus,
( x − 1)2 + 9
3
y
x − 1
y arcsin x 1 x 2 2 x 10
.
sin y x 1 x 12 9.
cos y 3
x 2 2 x 10
3
x 12 9
y arccos3
x 2 2 x 10.
68. Let
cosarcsin x h
r cos u r 2 x h2
r .
sin u x h
r , r
x h−
r x h− −( )22
u
u arcsin x h
r ,
8/10/2019 Chapter 4 Trig Other Book
http://slidepdf.com/reader/full/chapter-4-trig-other-book 70/103
Section 4.7 Inverse Trigonometric Functions 403
75.
Domain:
Range:
This is the graph of with a factor of 2.
−2 −1 1 2
π
π 2
x
y
f x arccos x
0 ≤ y ≤ 2
1 ≤ x ≤ 1
y 2 arccos x
76.
Domain:
Range:
This is the graph of
with a
horizontal stretch of a
factor of 2.
f x arcsin x
2 ≤ y ≤
2
2 ≤ x ≤ 2
1 2−2 x
π
π
y
−
y arcsin x
2 77.
Domain:
Range:
This is the graph of
shifted
one unit to the right.
g x arcsin x
2 ≤ y ≤
2
0 ≤ x ≤ 2
−1 1 2 3
π
π
y
x
−
f x arcsin x 1
78.
Domain:
Range:
This is the graph of
shifted
two units to the left.
y arccos t
0 ≤ y ≤
3 ≤ t ≤ 1
−4 −3 −2 −1t
π
ygt arccost 2 79.
Domain: all real numbers
Range:
This is the graph of
with a
horizontal stretch of a
factor of 2.
g x arctan x
2 < y <
2−4 −2 2 4
π
π
y
x
−
f x arctan 2 x
80.
Domain: all real numbers
Range:
This is the graph of
shifted
upward units. 2
y arctan x
0 < y ≤
−4 −2 2 4 x
π
y
f x
2 arctan x 81.
Domain:
Range: all real numbers
−2 1 2
π
y
v
1 − v2
v
y
1
1 ≤ v ≤ 1, v 0
hv tanarccos v 1 v2
v
74. If
then
2
u
x − 2
4 x x − 2
arccos x 2
2 arctan
4 x x 2
x 2.
cos u x 2
2,
arccos x 2
2 u,
8/10/2019 Chapter 4 Trig Other Book
http://slidepdf.com/reader/full/chapter-4-trig-other-book 71/103
8/10/2019 Chapter 4 Trig Other Book
http://slidepdf.com/reader/full/chapter-4-trig-other-book 72/103
Section 4.7 Inverse Trigonometric Functions 405
92. (a)
(b) When
When
arctan1200
750 1.01 58.0.
s 1200,
arctan300
750
0.38 21.8.
s 300,
arctans
750
tan s
75093.
(a)
(b) is maximum when feet.
(c) The graph has a horizontal asymptote at
As x increases, decreases.
0.
x 2
0
−0.5
6
1.5
arctan3 x
x 2 4
94. (a)
(b)
feeth 20 tan 20 11
17 12.94
tan hr h
20
r 1
240 20
arctan11
17 0.5743 32.9
tan 11
17 95.
(a)
(b)
h 50 tan 26 24.39 feet
tan 26 h
50
arctan20
41 26.0
tan 20
41
20 ft
41 ft
θ
96. (a)
(b)
arctan6
1 1.41 80.5
x 1 mile
arctan6
7 0.71 40.6
x 7 miles
arctan6
x
tan 6
x 97. (a)
(b)
x 12: arctan12
20 31.0
x 5: arctan 520
14.0
arctan x
20
tan x
20
98. False.
is not in the range of
arcsin1
2
6
arcsin x .5
6
99. False.
is not in the range of the arctangent function.
arctan 1
4
5
4
100. False.
is defined for all real x , but and require
Also, for example,
Since , but undefined.arcsin 1
arccos 1
2
0 arctan 1
4
arctan 1 arcsin 1
arccos 1.
1 ≤ x ≤ 1.arccos x arcsin x arctan x
8/10/2019 Chapter 4 Trig Other Book
http://slidepdf.com/reader/full/chapter-4-trig-other-book 73/103
8/10/2019 Chapter 4 Trig Other Book
http://slidepdf.com/reader/full/chapter-4-trig-other-book 74/103
8/10/2019 Chapter 4 Trig Other Book
http://slidepdf.com/reader/full/chapter-4-trig-other-book 75/103
8/10/2019 Chapter 4 Trig Other Book
http://slidepdf.com/reader/full/chapter-4-trig-other-book 76/103
8/10/2019 Chapter 4 Trig Other Book
http://slidepdf.com/reader/full/chapter-4-trig-other-book 77/103
8/10/2019 Chapter 4 Trig Other Book
http://slidepdf.com/reader/full/chapter-4-trig-other-book 78/103
Section 4.8 Applications and Models 411
7. Given:
B 90 72.08 17.92
A arccos 1652 72.08º
cos A 16
52
2448 12 17 49.48
a 522 162
b = 16
c = 52a
AC
Bb 16, c 52 8. Given:
8.03
arcsin 1.329.45 a
b = 1.32
c = 9.45
AC
Bsin B
b
c ⇒ B arcsin
b
c
cos A b
c ⇒ A arccos
b
c arccos
1.32
9.45 81.97
a c2 b2 87.5601 9.36
b 1.32, c 9.45
9. Given:
b
c = 430.5a
AC
B
12°15′
b 430.5 cos 1215 420.70
cos 1215 b
430.5
a 430.5 sin 1215 91.34
sin 1215 a
430.5
B 90 12 15 77 45
A 12 15 , c 430.5 10. Given:
a = 14.2
b
c
AC
B
65°12′
tan B ba
⇒ b a tan B 14.2 tan 65 12 30.73
cos B a
c ⇒ c
a
cos B
14.2
cos 65 12 33.85
A 90 B 90 65 12 24 48
B 65 12 , a 14.2
11.
12
b12
b
b
h
θ θ
h
1
24 tan 52
2.56 inches
tan h
12b ⇒ h
1
2b tan 12.
12
b12
b
b
h
θ θ
h
1
210 tan 18 1.62 meters
tan h
12b ⇒ h
1
2b tan
13.
12
b12
b
b
h
θ θ
h 1
246 tan 41 19.99 inches
tan h
12b ⇒ h
1
2 b tan 14.
12
b12
b
b
h
θ θ
h 1
211 tan 27 2.80 feet
tan h
12b ⇒ h
1
2b tan
8/10/2019 Chapter 4 Trig Other Book
http://slidepdf.com/reader/full/chapter-4-trig-other-book 79/103
8/10/2019 Chapter 4 Trig Other Book
http://slidepdf.com/reader/full/chapter-4-trig-other-book 80/103
Section 4.8 Applications and Models 413
26. (a) Since the airplane speed is
after one minute its distance travelled is 16,500 feet.
18°
16500a
a 16,500 sin 18 5099 ft
sin 18 a
16,500
275ft
sec60sec
min 16,500ft
min,
(b)
275s 10,000feet
18°
117.7 seconds
s 10,000
275(sin 18)
sin 18 10,000
275s
27.
x 4 sin 10.5 0.73 mile
10.5°
4 x sin 10.5
x
4
28.
Angle of grade:
Change in elevation:
2516.3 feet
21,120 sinarctan 0.12
y 21,120 sin
sin y
21,120
arctan 0.12 6.8
tan 12 x
100 x
100 x
12 = x y4 miles = 21,120 f eet
θ 29. The plane has traveled
N
S
EW
90052°
38°
a
b
cos 38 b
900 ⇒ b 709 miles east
sin 38 a
900 ⇒ a 554 miles north
1.5600 900 miles.
30. (a) Reno is miles N of Miami.
Reno is miles W of Miami.
(b) The return heading isN
S
EW
280°
10°
280.
2472 cos 10 2434
N
S
EW
100°
80°
10° Miami
Reno2472 mi
2472 sin 10 429
25.
arctan 950
26,400
2.06
tan 950
26,400
5 miles 5 miles5280 feet
1 mile 26,400 feet
5 miles
950feet
θ
θ
Not drawn to scale1200 feet 150 feet 400 feet 950 feet
8/10/2019 Chapter 4 Trig Other Book
http://slidepdf.com/reader/full/chapter-4-trig-other-book 81/103
8/10/2019 Chapter 4 Trig Other Book
http://slidepdf.com/reader/full/chapter-4-trig-other-book 82/103
Section 4.8 Applications and Models 415
39.
17,054 ft
a cot 16 a cot 57 55
6 ⇒ a 3.23 miles
cot 16 a cot 57 556a
tan 16 a
a cot 57 556
tan 16 a
x 556
57°16°
x 550
60
H
P1 P2
a
tan 57 a
x ⇒ x a cot 57
41.
arctan 5 78.7
tan
1 32
1
132
52
12
5
L2: 3 x y 1 ⇒ y x 1 ⇒ m
2 1
L1: 3 x 2 y 5 ⇒ y
3
2 x
5
2 ⇒ m
1
3
240.
5410 feet
h 17
1
tan 2.5
1
tan 9 1.025 miles
h
tan 2.5
h
tan 9 17
x h
tan 9 17
tan 9
h
x 17
x h
tan 2.5
x − 1717
2.5° 9°
x
h
Not drawn to scale
tan 2.5 h
x
42.
arctan9
7 52.1
arctanm
2 m
1
1 m2m
1 arctan 15 2
1 152
tan m
2 m
1
1 m2m
1
L2 x 5 y 4 ⇒ m
2 1
5
L1 2 x y 8 ⇒ m
1 2 43. The diagonal of the base has a length of
Now, we have
35.3.
arctan1
2
a
2a
θ
tan a
2a
1
2
a2 a2 2a.
44.
arctan 2 54.7º
θ
a
a 2
tan a 2
a 2 45.
Length of side:
36°
25
d
2d 29.4 inches
sin 36 d
25 ⇒ d 14.69
38.
Distance between towns:
d D
28°
28°
55°
55°10 km
T 2T 1
D d 18.8 7 11.8 kilometers
cot 28 D
10 ⇒ D 18.8 kilometers
cot 55 d
10 ⇒ d 7 kilometers
8/10/2019 Chapter 4 Trig Other Book
http://slidepdf.com/reader/full/chapter-4-trig-other-book 83/103
416 Chapter 4 Trigonometry
46.
25 inches
Length of side 2a 212.5
a 25 sin 30 12.5
sin 30 a
25
2530°
a 47.
y 2b 2 3r
2 3r
b 3r
2
b r cos 30
cos 30 b
r
br
r
30°
2
x
y
48.
Distance 2a 9.06 centimeters
4.53
a c sin 15 17.5 sin 15
sin 15 a
c
c 35
2 17.5
c
15°
a
49.
a 10
cos 35 12.2
cos 35 10
a
b 10 tan 35 7
tan 35 b
10
35° 35°
ba
10 10
10
10
10
10
50.
c 10.82 7.22 13 feet
b 6sin
7.2 feet
sin 6
b
90 33.7 56.3
f 21.6
2 10.8 feet
a 18
cos 21.6 feet
6
6bc
a f
θ φ
936
cos 18
a
arctan2
3 0.588 rad 33.7
tan 12
18
51.
Use since
Thus, d 4 sin t .
2
2 ⇒
d 0 when t 0.d a sin t
d 0 when t 0, a 4, period 2 52. Displacement at is
Amplitude:
Period:
d 3 sin t 3
2
6 ⇒
3
a 3
0 ⇒ d a sin t .t 0
53.
Use since
Thus, d 3 cos4
3t 3 cos4 t
3 .
2
1.5 ⇒
4
3
d 3 when t 0.d a cos t
d 3 when t 0, a 3, period 1.5 54. Displacement at is
Amplitude:
Period:
d 2 cos t 5
2
10 ⇒
5
a 2
2 ⇒ d a cos t .t 0
8/10/2019 Chapter 4 Trig Other Book
http://slidepdf.com/reader/full/chapter-4-trig-other-book 84/103
Section 4.8 Applications and Models 417
55.
(a) Maximum displacement amplitude
(b)
cycles per unit of time
(c)
(d) 8 t
2 ⇒ t
1
16
d 4 cos 40 4
4
Frequency
2
8
2
4
d 4 cos 8 t 56.
(a) Maximum displacement:
(b) Frequency: cycles per unit of time
(c)
(d) Least positive value for t for which
t
2
1
20
1
40
20 t
2
20 t arccos 0
cos 20 t 0
1
2 cos 20 t 0
d 0
t 5 ⇒ d 1
2 cos 100 1
2
2
20
2 10
a 1
2 1
2
d 1
2 cos 20 t
57.
(a) Maximum displacement amplitude
(b)
cycles per unit of time
(c)
(d) 120 t ⇒ t 1
120
d 1
16 sin 600 0
60
Frequency
2
120
2
1
16
d 1
16 sin 120 t
58.
(a) Maximum displacement:
(b) Frequency: cycles per unit of time
(c)
(d) Least positive value for t for which
t
792
1
792
792 t
792 t arcsin 0
sin 792 t 0
1
64 sin 792 t 0
d 0
t 5 ⇒ d 1
64 sin3960 0
2
792
2 396
a 1
64 1
64
d 1
64 sin 792 t
59.
2 264 528
264
2
Frequency
2
d a sin t 60. At buoy is at its high point
Returns to high point every 10 seconds:
Period:
d 7
4 cos
t
5
2
10 ⇒
5
a 74
Distance from high to low 2a 3.5
⇒ d a cos t .t 0,
8/10/2019 Chapter 4 Trig Other Book
http://slidepdf.com/reader/full/chapter-4-trig-other-book 85/103
418 Chapter 4 Trigonometry
61.
(a)
t
1
−1
y
π π 38 8
π 4
π 2
y 1
4 cos 16t , t > 0
(b) Period:
(c)1
4cos 16t 0 when 16t
2 ⇒ t
32
2
16
8
62. (a)
(c) L L1 L
2
2
sin
3
cos
(b)
The minimum length of the elevator is 7.0 meters.
(d)
From the graph, it appears that the minimum length is
7.0 meters, which agrees with the estimate of part (b).
−12
−2 2
12
0.1 23.0
0.2 13.1
0.3 9.9
0.4 8.43
cos 0.4
2
sin 0.4
3
cos 0.3
2
sin 0.3
3
cos 0.2
2
sin 0.2
3
cos 0.1
2
sin 0.1
L1 L
2 L
2 L
1
0.5 7.6
0.6 7.2
0.7 7.0
0.8 7.13
cos 0.8
2
sin 0.8
3
cos 0.7
2
sin 0.7
3
cos 0.6
2
sin 0.6
3
cos 0.5
2
sin 0.5
L1 L
2 L
2 L
1
63. (a) and (b)
Base 1 Base 2 Altitude Area
8 22.1
8 42.5
8 59.7
8 72.7
8 80.5
8 83.1
8 80.7
The maximum occurs when and is approximately
83.1 square feet.
60
8 sin 70º8 16 cos 70º
8 sin 60º8 16 cos 60º
8 sin 50º8 16 cos 50º
8 sin 40º8 16 cos 40º
8 sin 30º8 16 cos 30º
8 sin 20º8 16 cos 20º
8 sin 10º8 16 cos 10º
(c)
(d)
The maximum of 83.1 square feet occurs when
3 60.
0
900
100
641 cos sin
16 16 cos 4 sin
A 8 8 16 cos 8 sin
2
8/10/2019 Chapter 4 Trig Other Book
http://slidepdf.com/reader/full/chapter-4-trig-other-book 86/103
Section 4.8 Applications and Models 419
66. False. One period is the time for one complete cycle of
the motion.
68. Aeronautical bearings are always taken clockwise from
North (rather than the acute angle from a north-south line).
64. (a)
(c) Period:
This corresponds to the 12 months in a year. Since the
sales of outerwear is seasonal, this is reasonable.
2
6 12
Month (1 January)↔
Averagesales
(inmillionsofdollars)
t
S
2 4 121086
3
6
9
12
15
(b)
Shift:
Note: Another model is
The model is a good fit.
(d) The amplitude represents the maximum displacement from
average sales of 8 million dollars. Sales are greatest in
December (cold weather Christmas) and least in June.
S 8 6.3 sin t 6
2.
S 8 6.3 cos t 6
S d a cos bt
d 14.3 6.3 8
2
b 12 ⇒ b
6
t 2 4 121086
3
6
9
12
15
A v e r a g e s a l e s
( i n m i l l i o n s o f d o l l a r s )
Month (1 ↔ January)
S
a 1
214.3 1.7 6.3
65. False. Since the tower is not exactly vertical, a right
triangle with sides 191 feet and d is not formed.
67. No. N 24 E means 24 east of north.
69. passes through
y 4 x 6
y 2 4 x 4
y 2 4 x 1
y
x −1−2−3−4 1 2 3 4−1
1
2
3
5
6
7
1, 2m 4, 70. Linear equation through
y 12 x
16
b 16
0 16 b
0 12
13 b
y
x −1−2−3 2 3
−1
−2
−3
1
2
3
y 12 x b
13, 0m
12
71. Passes through and
y 4
5 x
22
5
y 6 45 x 8
5
y 6 4
5 x 2
y
x −1−2 1 2 3 4 5−1
1
2
3
4
6
7
m 2 6
3 2
4
5
3, 22, 6 72. Linear equation through and
y 4
3 x
1
3
y 2
3
4
3 x 1
4
4
3
1
34
y
x −1−2−3 2 3
−1
−2
−3
1
2
3
m 13 23
12 14
1
2,
1
31
4,
2
3
8/10/2019 Chapter 4 Trig Other Book
http://slidepdf.com/reader/full/chapter-4-trig-other-book 87/103
Review Exercises for Chapter 4
420 Chapter 4 Trigonometry
1. 0.5 radian 2. 4.5 radians
3.
(a)
(b) The angle lies in Quadrant II.
(c) Coterminal angles:
3
4 2
5
4
11
4 2
3
4
11
4
π
x
y
11
44.
(a)
(b) Quadrant I
(c)
2
9 2
16
9
2
9 2
20
9
x
y
2
9
π
2
95.
(a)
(b) The angle lies in Quadrant II.
(c) Coterminal angles:
4
3 2
10
3
4
3 2
2
3
−4
3
π
x
y
4
3
6.
(a)
(b) Quadrant I
(c)
23
3 2 17
3
23
3 8
3
x
−
y
23
3
π
23
3 7.
(a)
(b) The angle lies in Quadrant I.
(c) Coterminal angles:
70 360 290
70 360 430
70°
y
x
70 8.
(a)
(b) Quadrant IV
(c)
280 360 80
280 360 640
x
280°
y
280
8/10/2019 Chapter 4 Trig Other Book
http://slidepdf.com/reader/full/chapter-4-trig-other-book 88/103
Review Exercises for Chapter 4 421
9.
(a)
(b) The angle lies in Quadrant III.
(c) Coterminal angles:
110 360 470
110 360 250
−110°
x
y
110 10.
(a)
(b) Quadrant IV
(c)
405 360 45
405 720 315
x
−405°
y
405 11.
8.378 radians
8
3 radians
480 480 rad
180
12. 127.5
180 2.225 13.
3
16 radian 0.589 radian
33 45 33.75 33.75 rad
180
14. 196 77 196 77
60
180 3.443 15.
5 rad
7
5 rad
7
180
rad 128.571
16. 11
6
180
330.000 17. 3.5 rad 3.5 rad
180
rad 200.535
18. 5.7 180
326.586 19. radians
inchess r 2023
30 48.17
138 138
180
23
30 20.
meterss 11.52
11
3 meters
s r 11 60
180
60 60
180 radians
21. (a)
(b)
400 inches per minute
Linear speed 666
23 inches
1 minute
6623 radians per minute
Angular speed 33
132 radians
1 minute 22.
12.05 miles per hour
212.1 inches per second
67.5 inches per second
5 rads 13.5 inches
linear speed angular speed radius
23. radians
square inches A 1
2r 2
1
21822
3 339.29
120 120
180
2
324.
A 55.31 square millimeters
A 1
2 r 2
1
25
6 6.52
26. t 3
4, x , y 2
2, 2
2 25. corresponds to the point .1
2, 3
2 t 2
3
8/10/2019 Chapter 4 Trig Other Book
http://slidepdf.com/reader/full/chapter-4-trig-other-book 89/103
422 Chapter 4 Trigonometry
27. corresponds to the point 3
2,
1
2.t 5
6 28. t
4
3, x , y 1
2, 3
2
29. corresponds to the point
cot7
6
x
y 3tan
7
6
y
x
1
3 3
3
sec7
6
1
x
2 3
3cos
7
6 x
3
2
csc7
6
1
y
2sin7
6
y 1
2
3
2,
1
2.t 7
630. corresponds to the point
cot
4
x
y 1tan
4
y
x 1
sec
4
1
x 2cos
4 x
2
2
csc
4
1
y
2sin
4
y 2
2
2
2, 2
2 .t
4
31. corresponds to the point
cot2
3 x
y 3
3tan2
3 y
x 3
sec2
3 1
x 2cos2
3 x
1
2
csc2
3 1
y
2 3
3sin2
3 y 3
2
1
2,
3
2 .t 2
332. corresponds to the point
is undefined.
is undefined.cot 2 x
ytan 2
y
x 0
sec 2 1
x
1cos 2 x 1
csc 2 1
ysin 2 y 0
1, 0.t 2
33. sin11
4 sin
3
4
2
234. cos 4 cos 0 1 35. sin17
6 sin5
6 1
2
36. cos13
3 cos5
3 1
2 37. tan 33 75.3130 38. csc 10.5
1
sin 10.5 1.1368
39. sec12
5 1
cos12
5 3.2361 40. sin
9 0.3420
41.
cot adj
opp
5
4tan
opp
adj
4
5
cos adj
hyp
5
41
5 41
41 sec
hyp
adj
41
5
sin opp
hyp
4
41
4 41
41 csc
hyp
opp 41
4
opp 4, adj 5, hyp 42 52 41 42.
cot adj
opp
6
6 1tan
opp
adj
6
6 1
sec hyp
adj
6 2
6 2cos
adj
hyp
6
6 2 2
2
csc hyp
opp
6 2
6 2sin
opp
hyp
6
6 2 2
2
hyp 62 62 6 2
adj 6, opp 6
43.
cot adj
opp
4
4 3 3
3tan
opp
adj
4 3
4 3
cos adj
hyp
4
8
1
2 sec
hyp
adj
8
4 2
sin opp
hyp
4 3
8
3
2 csc
hyp
opp
8
4 3
2 3
3
adj 4, hyp 8, opp 82 42 48 4 3
8/10/2019 Chapter 4 Trig Other Book
http://slidepdf.com/reader/full/chapter-4-trig-other-book 90/103
Review Exercises for Chapter 4 423
44.
cot adj
opp
2 14
5
sec hyp
adj
9
2 14
9 14
28
csc hyp
opp
9
5
tan opp
adj
5
2 14
5 14
28
cos adj
hyp
2 14
9
sin opp
hyp
5
9
adj 92 52 2 14
opp 5, hyp 9 45.
(a)
(b)
(c)
(d) tan sin
cos
13
2 23
1
2 2 2
4
sec 1
cos
3
2 2
3 2
4
cos 2 2
3
cos 89
cos2 8
9
cos2 1 1
9
1
3
2
cos2 1
sin2 cos2 1
csc 1
sin 3
sin 1
3
46.
(a)
(b)
(c)
(d) csc
1
cot
2
1
1
16 17
4
cos 1
sec
1
17 17
17
sec 1 tan2 1 16 17
cot 1
tan
1
4
tan 4 47.
(a)
(b)
(c)
(d) tan sin
cos
14
154
1
15 15
15
sec 1
cos
4
15
4 15
15
cos 15
4
cos 15
16
cos2 15
16
cos2 1 1
16
1
42
cos2 1
sin2 cos2 1
sin 1
csc
1
4
csc 4
48.
(a)
(b) cot csc2 1 25 1 2 6
sin 1
csc
1
5
csc 5
(c)
(d) sec90 csc 5
tan 1
cot
1
2 6 6
12
49. tan 33 0.6494 51. sin 34.2 0.562150. csc 11 1
sin 11 5.2408
8/10/2019 Chapter 4 Trig Other Book
http://slidepdf.com/reader/full/chapter-4-trig-other-book 91/103
424 Chapter 4 Trigonometry
55.
kilometer or 71.3 meters
1 10'° x
Not drawn to scale
3. 5 k m
x 3.5 sin 1 10 0.07
sin 1 10 x
3.5 56.
x 25tan 52
19.5 feet
tan 52 25
x
52°
x
25
57.
cot x y 3
4tan y
x 4
3
sec r
x
5
3cos
x
r
3
5
csc r
y
5
4sin
y
r
4
5
x 12, y 16, r 144 256 400 20 58.
cot x
y
3
4tan
y
x
4
3
sec r
x
5
3
cos x
r
3
5
csc r
y
5
4sin
y
r
4
5
r 32 42 5
x , y 3, 4
59.
cot x
y
23
52
4
15tan
y
x
52
23
15
4
sec r
x 2416
23
241
4cos
x
r
23
2416
4
241
4 241
241
csc r
y 2416
52
2 241
30
241
15sin
y
r
52
2416
15
241
15 241
241
r 2
32
5
22
241
6
x 2
3, y
5
2
52. sec 79.3 1
cos 79.3 5.3860 53.
3.6722
cot 15 14 1
tan15 1460
54.
0.2045
cos 78 11 58 cos7811
60
58
3600
60.
cot x
y
103
23 5tan
y
x 23
103
1
5
sec
r
x
2 263
103
26
5cos
x
r
103
2 263
5 26
26
csc r
y
2 263
23 26sin
y
r
23
2 263
26
26
r 10
3 2
2
32
2 26
3
x , y 10
3,
2
3
8/10/2019 Chapter 4 Trig Other Book
http://slidepdf.com/reader/full/chapter-4-trig-other-book 92/103
Review Exercises for Chapter 4 425
61.
cot x
y
0.5
4.5
1
9tan
y
x
4.5
0.5 9
sec r
x
822
0.5
82cos x
r
0.5
822
82
82
csc r
y 822
4.5
82
9sin
y
r
4.5
822
9 82
82
r 0.52 4.52 20.5 82
2
x 0.5, y 4.5
62.
cot x y 0.3
0.4 3
4 0.75tan y
x 0.4
0.3 4
3 1.33
sec r
x
0.5
0.3
5
3 1.67cos
x
r
0.3
0.5
3
5 0.6
csc r
y
0.5
0.4
5
4 1.25sin
y
r
0.4
0.5
4
5 0.8
r 0.32 0.42 0.5
x , y 0.3, 0.4
64.
cot x
y
2 x
3 x
2
3
sec r
x 13 x
2 x
13
2
csc r
y 13 x
3 x
13
3
tan y
x
3 x
2 x
3
2
cos x
r
2 x
13 x
2 13
13
sin y
r
3 x
13 x
3 13
13
r 2 x 2 3 x 2 13 x
x , y 2 x , 3 x , x > 0 65. is in Quadrant IV.
cot 5 11
11
sec 6
5
csc r
y
6 11
11
tan y
x
11
5
cos x
r
5
6
sin y
r
11
6
r 6, x 5, y 36 25 11
sec 6
5, tan < 0 ⇒
63.
cot x
y x
4 x
1
4tan y
x
4 x
x 4
sec r
x 17 x
x 17cos
x
r
x
17 x 17
17
csc r
y 17 x
4 x
17
4sin
y
r
4 x
17 x
4 17
17
r x 2 4 x 2 17 x
x x , y 4 x
x , 4 x , x > 0
8/10/2019 Chapter 4 Trig Other Book
http://slidepdf.com/reader/full/chapter-4-trig-other-book 93/103
426 Chapter 4 Trigonometry
66.
is in Quadrant II.
cot 1
tan
5
2
sec 1
cos
3 5
5
tan sin
cos
2 5
5
cos
1
sin2
5
3
sin 1
csc
2
3
csc 3
2, cos < 0 67. is in Quadrant II.
cot 55
3
sec 8
55
8 55
55
csc 8
3
tan y
x
3
55
3 55
55
cos x
r
55
8
sin y
r
3
8
y 3, r 8, x 55
sin 3
8, cos < 0 ⇒
68.
is in Quadrant III.
cot
1
tan
4
5
csc 1
sin
41
5
sin 1 cos2 1 16
41
5 41
41
cos 1
sec
4 41
41
sec 1 tan2 1 25
16 41
4
tan 5
4
, cos < 0 69.
cot
x
y 2
21
2 21
21
sec r
x
5
2
5
2
csc r
y
5
21
5 21
21
tan y
x
21
2
sin y
r 21
5
sin > 0 ⇒ is in Quadrant II ⇒ y 21
cos x
r
2
5 ⇒ y2 21
70.
is in Quadrant IV.
cot 1
tan 3
tan sin
cos
3
3
sec 1
cos
2 3
3
cos 1 sin2 1 1
4 3
2
csc 1
sin 2
sin 2
4
1
2, cos > 0 71.
′θ
264°
x
y
264 180 84
264
8/10/2019 Chapter 4 Trig Other Book
http://slidepdf.com/reader/full/chapter-4-trig-other-book 94/103
Review Exercises for Chapter 4 427
72.
′θ
635°
x
y
85
635 720 85 73.
′θ x
y
6
5
π −
4
5
5
6
5 2
4
5
6
5 74.
′θ
x
y
17
3
π
3
6
3
17
3
18
3
3
75.
tan
3 3
cos 3 1
2
sin
3
3
2 76.
tan
4
2
22 1
cos 4 2
2
sin
4 2
277.
tan7
3 tan
3 3
cos7 3 cos
3 1
2
sin7
3 sin
3
3
2
78.
2
22 1
tan 5
4 tan
4
cos 5
4 cos
4
2
2
sin 5
4 sin
4
2
2 79.
tan 495 tan 45 1
cos 495 cos 45 2
2
sin 495 sin 45 2
280.
tan150 12
32 3
3
cos150 3
2
sin150 1
2
81.
tan240 tan 60 3
cos240 cos 60 1
2
sin240 sin 60 3
282.
tan315 22
22 1
cos315 2
2
sin315 2
2
83. sin 4
0.7568 84. tan 3
0.1425 85. sin
3.2 0.0584
86. cot4.8 1
tan4.8 0.0878 87. sec12
5 1
cos12
5 3.2361 88. tan25
7 4.3813
8/10/2019 Chapter 4 Trig Other Book
http://slidepdf.com/reader/full/chapter-4-trig-other-book 95/103
428 Chapter 4 Trigonometry
89.
2
1
−2
x
− π π 3
22
y
Period: 2
Amplitude: 1
y sin x 91.
Amplitude: 5
Period:
−6
−2
2
4
6
x
π 6
y
2
25 5
f x 5 sin2 x
590.
Amplitude: 1
Period:
x
2
−1
−2
2π π π −
2
y cos x
92.
Amplitude: 8
Period:
−8
−6
−4
8
x
y
π 8π 4
2 14
8
f x 8 cos x
4 93.
Shift the graph of
two units upward.
4
3
2
−1
−2
x
π π π 2
y
−
y sin x
y 2 sin x 94.
Amplitude: 1
Period:
321−2−3 x
−1
−2
−3
−5
−6
−1
2
2
y 4 cos x
96.
Amplitude: 3
Period:
−4
−3
1
2
3
4
t π
y
2
gt 3 cost 97.
(a)
(b)
264 cycles per second.
f 1
1264
y 2 sin528 x
2
b
1
264 ⇒ b 528
a 2,
y a sin bx
98. (a)
0
14
12
22
S t 18.09 1.41 sin t
6 4.60
95.
Amplitude:
Period:
−4
−3
−2
−1
1
3
4
t π
y
2
5
2
gt 5
2 sint
(b)
so this is expected.
(c) Amplitude: 1.41
The amplitude represents the maximum change in the time
of sunset from the average time .d 18.09
12 months 1 year,
Period 2
6 26 12
8/10/2019 Chapter 4 Trig Other Book
http://slidepdf.com/reader/full/chapter-4-trig-other-book 96/103
8/10/2019 Chapter 4 Trig Other Book
http://slidepdf.com/reader/full/chapter-4-trig-other-book 97/103
430 Chapter 4 Trigonometry
119. arccos 0.324 1.24 radians118. cos1 3
2
6 120. radiansarccos0.888 2.66
122. radianstan18.2 1.45121. tan1 1.5 0.98 radian
127.
Use a right triangle. Let
then
and cos 45.
tan 34 arctan
34
θ
5
4
3
cosarctan34
45
126.
−1.5 1.5
2
−
2
f x arcsin 2 x 125.
−4 4
2
−
2
f x arctan x
2 tan1 x
2
123.
−1.5 1.5
−
f x 2 arcsin x 2 sin1 x 124.
−1.5 1.5
3
0
y 3 arccos x
128. Let
tanarccos35 tan u
43
4
u
3
5
u arccos35.
129.
Use a right triangle. Let
then and sec 135 .tan
125
arctan125
θ
5
1213
secarctan125
135 130. Let
cot arcsin1213 cot u
512
u
−1213
5u arcsin1213.
131. Let Then
and
2
y
x
4 − x 2
tan y tanarccos x
2 4 x 2
x .cos y
x
2
y arccos x
2. 132.
x −11
θ
12 − ( x − 1)2
sec 1
x 2 x
cos 12 x 12 x 2 x sin
x
1
arcsin x 1 ⇒
2 ≤ ≤
2
secarcsin x 1
8/10/2019 Chapter 4 Trig Other Book
http://slidepdf.com/reader/full/chapter-4-trig-other-book 98/103
Review Exercises for Chapter 4 431
133.
arctan70
30 66.8
tan 70
30 134.
h 25 tan 21 9.6 feeth
25
21°
tan 21 h
25
135.
The distance is 1221 miles and the bearing is 85.6.
sec 4.4 D
1217 ⇒ D 1217 sec 4.4 1221
tan 93
1217 ⇒ 4.4
sin 25 d
4
810 ⇒ d
4 342
cos 48 d
3
650 ⇒ d
3 435
cos 25 d
2
810 ⇒ d
2 734
sin 48 d
1
650
⇒ d 1 483
48°
48°
65°25°
d 3d 4
d d 1 2
810650
D
B
C A θ
N
S
WE
d 1 d
2 1217
d 3 d
4 93
136. Amplitude:
Period: 3 seconds
d 0.75 cos2 t
3
b 2
3
a 0.75
d a cos bt
1.5
2 0.75 inches 137. False. The sine or cosine functions
are often useful for modeling
simple harmonic motion.
138. True. The inverse sine,
is defined where
and
2 ≤ y ≤
2.
1 ≤ x ≤ 1
y arcsin x ,
139. False. For each there
corresponds exactly one
value of . y
140. False. The range of arctan is
so arctan1
4.
2,
2,
141.
Matches graph d
Period: 2
Amplitude: 3
y 3 sin x
142. matches graph (a).
Period:
Amplitude: 3
2
y 3 sin x 143.
Matches graph bPeriod: 2
Amplitude: 2
y 2 sin x 144. matches graph (c).
Period:
Amplitude: 2
4
y 2 sin x
2
145. is undefined at the zeros of since sec 1
cos .g cos f sec
8/10/2019 Chapter 4 Trig Other Book
http://slidepdf.com/reader/full/chapter-4-trig-other-book 99/103
432 Chapter 4 Trigonometry
Problem Solving for Chapter 4
1. (a)
revolutions
radians or
(b) s r 47.255.5 816.42 feet
990 11
4 2 11
2
132
48
11
4
8:57 6:45 2 hours 12 minutes 132 minutes 2. Gear 1:
Gear 2:
Gear 3:
Gear 4:
Gear 5:24
19
360 454.737 7.94 radians
40
32360 450
5
2 radians
24
22360 392.727 6.85 radians
24
26360 332.308 5.80 radians
24
32360 270
3
2 radians
3. (a)
(b)
x 3000
tan 39 3705 feet
tan 39 3000
x
d 3000
sin 39 4767 feet
sin 39 3000
d (c)
w 3000 tan 63 3705 2183 feet
3000 tan 63 w 3705
tan 63 w 3705
3000
146. (a) (b) tan
2 cot 0.1 0.4 0.7 1.0 1.3
0.27760.64211.18722.36529.9666cot
0.27760.64211.18722.36529.9666tan
2
147. The ranges for the other four trigonometric functions
are not bounded. For the range
is For the range is
, 1 1, . y sec x and y csc x ,, .
y tan x and y cot x ,
148.
(a) A is changed from The displacement is
increased.
(b) k is changed from The friction damps the
oscillations more rapidly.
(c) b is changed from 6 to 9: The frequency of oscilla-
tion is increased.
110 to
13 :
15 to
13 :
y Aekt cos bt 15et 10 cos 6t
149.
(a)
As r increases, the area function increases more rapidly.
0
0 6
4
A s
s r 0.8 0.8r , r > 0
A 12 r 20.8 0.4r 2, r > 0
A 12 r 2 , s r
150. Answers will vary.
(b)
0
3
30
As
0
s 10 , > 0
A 12 102 50 , > 0
8/10/2019 Chapter 4 Trig Other Book
http://slidepdf.com/reader/full/chapter-4-trig-other-book 100/103
Problem Solving for Chapter 4 433
5. (a)
h is even.
(b)
h is even.
−1
−2
3
2
h x sin2 x
−1
−2
3
2
h x cos2 x
7. If we alter the model so that we can use
either a sine or a cosine model.
For the cosine model we have:
For the sine model we have:
Notice that we needed the horizontal shift so that the sine
value was one when .
Another model would be:
Here we wanted the sine value to be 1 when t 0.
h 51 50 sin8 t 3
2 t 0
h 51 50 sin8 t
2h 51 50 cos8 t
b 8
d 1
2max min
1
2101 1 51
a 1
2max min
1
2101 1 50
h 1 when t 0, 8.
(a)
(b)
This is the time between heartbeats.
(c) Amplitude: 20
The blood pressure ranges between
and
(d) Pulse rate
(e) Period
64 60
2 b ⇒ b
64
60 2
32
15
60
64
15
16 sec
60 secmin34 secbeat
80 beatsmin
100 20 120.
100 20 80
Period 2
8 3
6
8
3
4 sec
0
70
5
130
P 100 20 cos8 3
t
6. Given: f is an even function and g is an odd function.
(a)
since f is even
Thus, h is an even function.(b)
since g is odd
Thus, h is an even function.
Conjecture: The square of either an even function or an
odd function is an even function.
h x
g x 2
g x 2
h x g x 2
h x g x 2
h x
f x 2
h x f x 2
h x f x 2
4. (a) are all similar triangles since they all have the same angles. is part of
all three triangles and Thus,
(b) Since the triangles are similar, the ratios of corresponding sides are equal.
(c) Since the ratios: it does not matter which triangle is used to calculate sin A.
Any triangle similar to these three triangles could be used to find sin A. The value of sin A would not change.
(d) Since the values of all six trigonometric functions can be found by taking the ratios of the sides of a right triangle,
similar triangles would yield the same values.
opp
hyp
BC
AB
DE
AD
FG
AF sin A
BC
AB
DE
AD
FG
AF
B D F .C E G 90.
A ABC , ADE , and AFG
8/10/2019 Chapter 4 Trig Other Book
http://slidepdf.com/reader/full/chapter-4-trig-other-book 101/103
434 Chapter 4 Trigonometry
9. Physical (23 days):
Emotional (28 days):
Intellectual (33 days):
(a)
(b) Number of days since birth until September 1, 2006:
5 11 31 1
20 years leap years remaining August days day in
July days September
All three drop early in the month, then peak toward the middle of the month, and drop again
toward the latter part of the month.
(c) For September 22, 2006, use
I 0.945
E 0.901
P 0.631
t 7369.
−2
7349 7379
2
P
I
E
t 7348
t 365 20
−2
7300 7380
2
P I E
I sin2 t
33, t ≥ 0
E sin2 t
28, t ≥ 0
P sin2 t
23, t ≥ 0
10.
(a)
(b) The period of
The period of
(c) is periodic since the sine
and cosine functions are periodic.
h x A cos x B sin x
g x is .
f x is 2 .
−6
−
6
g
f
g x 2 cos 2 x 3 sin 4 x
f x 2 cos 2 x 3 sin 3 x 11. (a) Both graphs have a period of 2 and intersect when
They should also intersect when
and
(b) The graphs intersect when
(c) Since and
the graphs will intersect again at these values. Therefore
f 13.35 g4.65.
4.65 5.35 5213.35 5.35 42
x 5.35 32 0.65.
x 5.35 2 7.35. x 5.35 2 3.35
x 5.35.
8/10/2019 Chapter 4 Trig Other Book
http://slidepdf.com/reader/full/chapter-4-trig-other-book 102/103
Problem Solving for Chapter 4 435
12. (a) is true since this is a two period
horizontal shift.
(b) is not true.
is a horizontal translation of
is a doubling of the period of
(c) is not true.
is a horizontal
translation of by half a period.
For example, sin1
2 2 sin1
2 .
f 1
2t
f 1
2 t c f 1
2t
1
2 c
f 1
2 t c f 1
2t
f t . f 12
t
f t . f t 1
2c
f t 1
2c f 1
2t
f t 2c f t 13.
(a)
(b)
(c) feet
(d) As you more closer to the rock, decreases, which
causes y to decrease, which in turn causes d todecrease.
1
d y x 3.46 1.71 1.75
tan 1
y
2 ⇒ y 2 tan 60 3.46 feet
tan 2
x
2 ⇒ x 2 tan 40.52 1.71 feet
2 40.52
sin 2
sin 1
1.333
sin 601.333
0.6497
sin 1
sin 2 1.333
2 ft
x y
d
θ
θ 1
2
14.
(a)
The graphs are nearly the same for 1 < x < 1.
−2
2
2−
2
arctan x x x 3
3
x 5
5
x 7
7
(b)
The accuracy of the approximation improved slightly by
adding the next term x 99.
−2
2
2−
2
8/10/2019 Chapter 4 Trig Other Book
http://slidepdf.com/reader/full/chapter-4-trig-other-book 103/103
Chapter 4 Practice Test
436 Chapter 4 Trigonometry
1. Express 350° in radian measure. 2. Express in degree measure.5 9
3. Convert to decimal form.135 14 12 4. Convert form.22.569 to D M S
5. If use the trigonometric identities to find tan . cos 23, 6. Find given .sin 0.9063
7. Solve for in the figure below.
20°
35
x
x 8. Find the reference angle for . 6 5
9. Evaluate .csc 3.92 10. Find .sec given that lies in Quadrant III and tan 6
11. Graph y 3 sin x
2. 12. Graph y 2 cos x .
13. Graph y tan 2 x . 14. Graph . y csc x
4
15. Graph using a graphing calculator. y 2 x sin x , 16. Graph using a graphing calculator. y 3 x cos x ,
17. Evaluate .arcsin 1 18. Evaluate arctan3.
19. Evaluate sinarccos
4
35. 20. Write an algebraic expression for .cosarcsin
x
4For Exercises 21–23, solve the right triangle.
B