Download 1.2 Exercises

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Exercises
1. Explain the difference between a segment and a ray.
3. Concept Check
What angle is its own complement?
2. What part of a complete revolution is an angle of 45°?
4. Concept Check
What angle is its own supplement?
Find the measure of the smaller angle formed by the hands of a clock at the following times.
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Find the measure of each angle in Exercises 7-12. See Example 1.
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9.
22
The Trigonometric Functions
Chapter1
+
10. supplementary
angles with measures
10m
11. supplementary
angles with measures
6x - 4 and 8x -
12. complementary
angles with measures 9z
Concept Check
Use the concepts presented
tion to answer each question.
13. If an angle measures
x
O
,
+
7 and 7m
+
3 degrees
12 degrees
6 and 3z degrees
15. If a positive angle has measure
in this sec-
how can we represent
its
16. If a negative angle has measure X between 0° and
- 60°, how can we represent the first positive angle
complement?
O
14. If an angle measures
x
O
,
how can we represent
its
coterminal
supplement?
Perform each calculation.
17.62° 18'
with it?
See Example 2.
+ 21° 41'
18. 75° 15'
+ 83° 32'
19.71° 18' - 47° 29'
22. 180° - 124° 51'
20. 47° 23' - 73° 48'
21. 90° - 51° 28'
23. 90° - 72° 58' II"
24. 90° - 36° 18' 47"
Convert each angle measure
Example 3.
to decimal
degrees.
Use a calculator,
and round to the nearest thousandth
25. 20° 54'
26. 38° 42'
27. 91° 35' 54"
28. 34° 51' 35"
29. 274° 18' 59"
30. 165° 51' 9"
Convert each angle measure to degrees, -minutes, and seconds.
Use a calculator
as necessary.
32. 59.0854°
33. 89.9004°
34. 102.3771°
35. 178.5994°
36. 122.6853°
about the degree symbol (0) in the manual for
your graphing calculator. How is it used?
Find the angle of smallest positive measure coterminal
~38.
of a degree. See
See Example 3.
31. 31.4296°
ru 37. Read
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between 0° and 60°,
the first negative angle cotermi-
how can we represent
nal with it?
XO
Show that 1.21 hours is the same as 1 hour, 12 minutes, and 36 seconds. Discuss the similarity between
converting hours, minutes, and seconds to decimal
hours and converting degrees, minutes, and seconds to
decimal degrees.
with each angle. See Example 4.
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39. -40°
40. -98°
41. -125°
42. -203°
'C'>
43. 539°
44. 699°
45. 850°
46. 1000°
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Give an expression
-...J
47. 30°
that generates
all angles coterminal
48. 45°
with each angle. Let n represent any integer.
49. 135°
50. 270°
51. -90°
~ 53. Explain why the answers to Exercises 50 and 51 give the same set of angles.
54. Concept Check
A. 360°
+ r"
Which two of the following
B. r" - 360°
are not coterminal
C. 360° - r"
ru Consider
the function Y1 = 360((Xj360) - int(Xj360))
specified on a graphing calculator. (Note: The value of
int(x) is the largest integer less than or equal to x. With
some calculators, int is found in the MATH menu.} The
screen here shows that for X = 908 and X = -75, the
junction returns the smallest possible positive measure
coterminal with the angle. See Example 4. Use Y1 to do the
following.
55. Rework Exercise 39 with a graphing calculator.
56. Rework Exercise 40 with a graphing calculator.
D. r"
with rO?
+ 180°
V1(908)
V1 (-75)
188
285
52. -135°
1.2 Angles
23
Concept Check Sketch each angle in standard position. Draw an arrow representing the correct amount of rotation. Find
the measure of two other angles, one positive and one negative, that are coterminal with the given angle. Give the quadrant
of each angle.
57. 75°
5S. 89°
59. 174°
60. 234°
61. 300°
62. 512°
63. -61°
64. -159°
Concept Check Locate each point in a coordinate system. Draw a ray from the origin through the given point. Indicate with
an arrow the angle in standard position having smallest positive measure. Then find the distance rfrom the origin to the point,
using the distance formula of Section 1.1.
65. (-3, -3)
6S.
66. (-5,2)
(\13,1)
69.
(-2,2\13)
Solve each problem. See Example 5.
71. Revolutions of a Windmill A windmill makes 90
revolutions per minute. How many revolutions does it
make per second?
72. Revolutions of a Turntable A turntable in a shop
makes 45 revolutions per minute. How many revolutions does it make per second?
73. Rotating Tire A tire is rotating 600 times per minute.
Through how many degrees does a point on the edge
of the tire move in 1/2 second?
67. (-3, -5)
70. (4\13, -4)
grees that a point on the edge of the propeller will rotate in I second.
75. Rotating Pulley A pulley rotates through 75° in one
minute. How many rotations does the pulley make in
an hour?
76. Surveying One student in a surveying class measures an angle as 74.25°, while another student measures the same angle as 74° 20'. Find the difference
between these measurements, both to the nearest
minute and to the nearest hundredth of a degree.
74. Rotating Airplane Propeller An airplane-propeller
rotates 1000 times per minute. Find the number of deAngle of a Star Refer to thefigure and table given
in the chapter introduction. For each star in Exercises 77 and 78, find the measure of the other acute
angle in thefigure, 90° - e, using the values from the table.
77. Alpha Centauri
79.
7S.61 Cygni
Viewing Field of a Telescope Due to Earth's
rotation, celestial objects like the moon and the
stars appear to move across the sky, rising in the
east and setting in the west. As a result, if a telescope on
Earth remains stationary while viewing a celestial object, the object will slowly move outside the viewing
field of the telescope. For this reason, a motor is often
attached to telescopes so that the telescope rotates at the
same rate as Earth. Determine how long it should take
the motor to turn the telescope through an angle of
1 minute in a direction perpendicular to Earth's axis.
SO. Angle Measure of a Star on the American Flag Determine the measure of the angle in each point of the
five-pointed star appearing on the American flag.
(Hint: Inscribe the star in a circle, and use the following theorem from geometry: An angle whose vertex lies
on the circumference of a circle is equal to half the central angle that cuts off the same arc. See the figure.)