Harvard University Trigonometry Exercises

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1.1
142 Chapter 1 Trigonometry
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EXERCISES
VOCABULARY: Fill in the blanks
1.
means “measurement of triangles
2. An
is determined by rotating a ray about its endpoint.
3. Two angles that have the same initial and terminal sides are
4. One
is
5. Angles that measure between 0 and 7/2 are
the measure of a central angle that intercepts an arc equal to the radius of the circle.
angles, and angles that measure between 2 and
angles
angles, whereas two positive angles that have a sum
6. Two positive angles that have a sum of /2 are
of rare
angles.
7. The angle measure that is equivalent to a rotation of of a complete revolution about an angle’s vertex is
one
8. 180 degrees
radians
speed of a particle
9. The
is the ratio of central angle to time traveled.
– speed of a particle is the ratio of are length to time traveled, and the
10. The area of a sector of a circle with radius r and central angle 8, where is measured in radians, is given by
formula
SKILLS AND APPLICATIONS
(b) – 3
In Exercises 11-16, estimate the angle to the nearest one-half
radian.
11.
12.
11
25. (a)
6
26. (a) 4
(b) 7
In Exercises 27-30, determine two coterminal angles for
positive and one negative) for each angle. Give your anses
in radians.
(b)
27. (a)
14.
13.
8
6
0
15.
16.
28. (a)
(b)
?
6
?
0
In Exercises 17-22, determine the quadrant in which each
angle lies. (The angle measure is given in radians.)
5
11
97
17. (a)
18. ()
(b)
4
8
8
57
11
19. (a)
20. (
(b)
6
9
21. (a) 3.5 (b) 2.25 22. (a) 6.02 (b) -4.25
O
29. (a) 0 =
(b) =
In Exercises 23-26, sketch each angle in standard position.
30. (a) 8 =
9.
4
2.
23. (a) (b)-2.5 24. (a)
(b) 0 =
(a
ST
(b)
15
In Exercises 31-34, find if possible) the complement and
Section 1.1
Radian and Degree Measure
143
supplement of each angle,
31. (a) =/3 (b) 4/4
(b) 2
52. (a) = -390
32. (a) /12 (6) 11/12
34. (a) 3 (6) 1.5
(b) = 2304
33. (a) 1
In Exercises 35-40, estimate the number of degrees in the
angle. Use a protractor to check your answer.
36.
(b) 85
35.
In Exercises 53-56, find of possible) the complement and
supplement of each angle.
53. (a) 18
54, (a) 46
(b) 93
55. (a) 150? (b) 79? 56. (a) 130
(b) 170
In Exercises 57-60, rewrite each angle in radian measure as a
multiple of m. (Do not use a calculator.)
57. (a) 30 (b) 45?
58. (a) 315″ (b) 120
59. (a) – 20
(b) -60? 60. (a) – 270 (b) 144
In Exercises 61-64, rewrite each angle in degree measure.
(Do not use a calculator.)
37.
38.
61. (a)
37
2
39.
(b)
62. (a)
7
12
40.
TT
(b)
b)
9
75
6
7
3
63. (a)
5
4
(b)
64. (a)
11
6
(b)
34
15
In Exercises 41-44, determine the quadrant in which each
angle lies.
41. (a) 130?
(b) 285
42. (a) 8.3?
(b) 257? 30′
43. (a) -132? 50′
(b) -336
44. (a) -260?
(b) -3.4?
In Exercises 65-72, convert the angle measure from degrees
to radians. Round to three decimal places.
65. 45
66. 87.4
67. – 216.35
68.-48.27
69. 532
70. 345
71. -0,83
72, 0.54
In Exercises 45-48, sketch each angle in standard position.
45. (a) 90? (b) 180?46. (a) 270? (b) 120?
47. (a) -30? (b)- 135?
48. (a) – 750? (b)-600?
In Exercises 73-80, convert the angle measure from radians
to degrees. Round to three decimal places.
73./7
74. 5/11
05. 157/8
76. 137/2
77.-4.2
780 4.8
79. – 2
80.-0.57
In Exercises 49-52, determine two coterminal angles (one
positive and one negative) for each angle. Give your answers
in degrees.
49. (a)
(b)
90?
1 0 = 45
90
In Exercises 81-84, convert each angle measure to decimal
degree form without using a calculator. Then check your
answers using a calculator.
(811. (a) 54? 45 (b) – 128? 30
82. (a) 245? 10′ (b) 2 12
83. a) 85? 18’30” (b) 330? 25″
84) (a) – 1359 36″ (b)-408? 16’20”
180?
0
180″-
-0
8 = -36″
270
270
50. (a)
90
(b)
90
9 = 120?
0-4200
0
180
0
180
F
In Exercises 85-88, convert each angle measure to degrees,
minutes, and seconds without using a calculator. Then check
your answers using a calculator.
85. (a) 240.6 (b) — 145.8?
86. (a) -345.12? (b) 0.45
(87 (a) 2.5 (b) – 3.58
88. (a) -0.36? (b) 0.799
270
270
51. (a) 6 = 240?
(b) 8 = – 180?
144
Chapter 1
Trigonometry
Ciry
106. San Francisco, California
Seattle, Washington
Latitude
37? 47′ 36″ N
47′ 37’18N
la Exercises 89-92, find the length of the are on a circle of
radius intercepted by a central angle
Radius
Central Angle 8
89, 15 inches
120
o feet
60
91. 3 meters
150
92. 20 centimeters
45
113. LINE
a DV
moto
preet
depe
(a)
(b)
107. DIFFERENCE IN LATITUDES Assuming that
is a sphere of radius 6378 kilometers. W
difference in the latitudes of Syracuse. Now
and Annapolis, Maryland, where Syracuse
450 kilometers due north of Annapolis?
jos, DIFFERENCE IN LATITUDES Assuming that
is a sphere of radius 6378 kilometers, what
difference in the latitudes of Lynchburg. Visa
Myrtle Beach, South Carolina, where Lynch
about 400 kilometers due north of Myrtle Beach
109. INSTRUMENTATION The pointer on a volte
6 centimeters in length (see figure). Find the
through which the pointer rotates when it
2.5 centimeters on the scale,
114. AN
an
In Exercises 93-96, find the radian measure of the central
angle of a circle of radius r that intercepts an arc of length s.
Radius
Are Length
18 inches
94. 14 feet
8 feet
95. 25 centimeters 10.5 centimeters
96 80 kilometers 150 kilometers
????
93) 4 inches
pu
(
115.
In Exercises 97-100, use the given arc length and radius to
find the angle (in radians).
97.
98.
25
9
116.
TO
6 cm
100.
75
FIGURE FOR 109
FIGURE TOR 110
60
11
1
In Exercises 101-104, find the area of the sector of the circle
with radius r and central angle 8.
Radius
Central Angle e
101. 6 inches
7/3
602. 12 millimeters
103. 2.5 feet
2259
104. 1.4 miles
330
110. ELECTRIC HOIST An electric hoist is being used
lift a beam (see figure). The diameter of the drums
the hoist is 10 inches, and the beam must be
2 feet. Find the number of degrees through which te
drum must rotate
111. LINEAR AND ANGULAR SPEEDS A circular po
saw has a 7-inch-diameter blade that rotate
5000 revolutions per minute.
(a) Find the angular speed of the saw blade in radius
per minute.
(b) Find the linear speed (in feet per minute) of oned
the 24 cutting teeth as they contact the wood be
cut.
112. LINEAR AND ANGULAR SPEEDS A carousel
a 50-foot diameter makes 4 revolutions per minute
(a) Find the angular speed of the carousel in tale
per minute.
(b) Find the linear speed (in feet per minute) of
platform rim of the carousel.
DISTANCE BETWEEN CITIES In Exercises 105 and 106,
find the distance between the cities. Assume that Earth is a
sphere of radius 4000 miles and that the cities are on the
same longitude (one city is due north of the other).
City
Latitude
105 Dallas, Texas
32? 47’39” N —
Omaha, Nebraska
41? 15’50” N

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