Download 426 16–1 Radian Measure Angle Conversions

satellites
satellites
at at
DRTE.
DRTE.
Dr.Dr.
Franklin
Franklin
moved
moved
on on
to to
thethe
Department
Department
of of
Communications
Communications
in 1969
in 1969
and
and became
became
the project
the project
manager
manager
for the
for Hermes
the Hermes
communications
communications
satellite
satellite
in the
in early
the early
1970s.
1970s.
He remained involved in developing space communications technology, both at home and
internationally, and in the mid-1980s helped draft a submission for the establishment of the
Canadian Space Agency.
Following a stint in the late 1980s as a visiting professor of electrical engineering in New
Zealand, Dr. Franklin spent two years as chief scientist at Spar Aerospace, the company that
designed and built the Canadarm. He was appointed to the Order of Canada in 1990.
16–1
c,
Ra
diu
s,
r
Ar
al
ntr
Ce le, g
an
S
Radian Measure
A central angle is one whose vertex is at the centre of the circle (Fig. 16–1). An arc is
a portion of the circle. In Chapter 6 we introduced the radian: If an arc is laid off along
a circle with a length equal to the radius of the circle, the central angle subtended by
this arc is defined as one radian, as shown in Fig. 16–2.
Angle Conversion
An arc having a length equal to the radius of the circle subtends a central angle of
one radian; an arc having a length of twice the radius subtends a central angle of two
radians; and so on. Thus an arc with a length of 2 times the radius (the entire circumference) subtends a central angle of 2 radians. Therefore 2 radians is equal to
1 revolution, or 360. This gives us the following conversions for angular measure:
FIGURE 16–1
Angle
Conversions
1 rev 360 2 rad
117
We convert angular units in the same way that we converted other units in Chapter 1.
One
radian
◆◆◆
Example 1: Convert 47.6 to radians and revolutions.
Solution: By Eq. 117,
2 rad
47.6 p q 0.831 rad
360
r
r
Note that 2 and 360 are exact numbers, so we keep the same number of digits in our
answer as in the given angle. Now converting to revolutions, we obtain
r
1 rev
47.6 p q 0.132 rev
360
FIGURE 16–2
◆◆◆
Example 2: Convert 1.8473 rad to degrees and revolutions.
Solution: By Eq. 117,
If we divide 360 by 2 we get
1 rad ⬇ 57.3
Some prefer to use Eq. 117 in
this approximate form instead of
the one given.
426
◆◆◆
360
1.8473 rad p q 105.84
2 rad
and
1 rev
1.8473 rad p q 0.294 01 rev
2 rad
◆◆◆
Section 16–1
◆◆◆
◆
427
Radian Measure
Example 3: Convert 1.837 520 rad to degrees, minutes, and seconds.
Solution: We first convert to decimal degrees.
360
1.837 520 rad p q 105.2821
2 rad
Converting the decimal part (0.2821) to minutes, we obtain
60
0.2821 p q 16.93
1
Converting the decimal part of 16.93 to seconds yields
60
0.93 p q 56
1
So 1.837 520 rad 1051656.
◆◆◆
Radian Measure in Terms of ␲
Not only can we express radians in decimal form, but it is also very common to express radian
measure in terms of . We know that 180 equals radians, so
90 rad
2
45° rad
4
15° rad
12
and so on. Thus to convert an angle from degrees to radians in terms of , multiply the angle
by ( rad/180), and reduce to lowest terms.
◆◆◆
Example 4: Express 135 in radian measure in terms of .
Solution:
135
3
rad
135 p q rad rad
180
4
180
◆◆◆
Students sometimes confuse the decimal value of with the
degree equivalent of radians.
What is the value of ? 180 or 3.1416 . . .?
Remember that the approximate decimal value of is always
Common
Error
⬵ 3.1416
but that radians converted to degrees equals
radians 180
Note that we write “radians” or “rad” after when referring
to the angle, but not when referring to the decimal value.
To convert an angle from radians to degrees, multiply the angle by (180/ rad). Cancel
the in numerator and denominator, and reduce.
◆◆◆
Example 5: Convert 7/9 rad to degrees.
Solution:
7
9
p
180 q
rad
7(180)
rad deg 140
9
◆◆◆
428
Chapter 16
◆
Radian Measure, Arc Length, and Circular Motion
Trigonometric Functions of Angles in Radians
We use a calculator to find the trigonometric functions of angles in radians just as we did for
angles in degrees. However, we first switch the calculator into radian mode. Consult your calculator manual for how to switch to this mode.
◆◆◆
Example 6: Use your calculator to verify the following to four decimal places:
(a) sin 2.83 rad 0.3066
(b) cos 1.52 rad 0.0508
(c) tan 0.463 rad 0.4992
◆◆◆
To find the cotangent, secant, or cosecant, we use the reciprocal relations, just as when
working in degrees.
◆◆◆
Example 7: Find sec 0.733 rad to three decimal places.
Solution: We put the calculator into radian mode. The reciprocal of the secant is the cosine, so
we take the cosine of 0.733 rad and get
cos 0.733 rad 0.7432
Then taking the reciprocal of 0.7432 gives
1
1
sec 0.733 1.346
cos 0.733 0.7432
◆◆◆
◆◆◆
Example 8: Use your calculator to verify the following to four decimal places:
(a) csc 1.33 rad 1.0297
(b) cot 1.22 rad 0.3659
(c) sec 0.726 rad 1.3372
◆◆◆
The Inverse Trigonometric Functions
The inverse trigonometric functions are found the same way as when working in degrees. Just
be sure that your calculator is in radian mode.
As with the trigonometric functions, some calculators require that you press the function
key before entering the number, and some require that you press it after entering the number.
◆◆◆
Example 9: Use your calculator to verify the following, in radians, to four decimal places:
(a) arcsin 0.2373 0.2396 rad
(b) cos1 0.5152 1.0296 rad
(c) arctan 3.246 1.2720 rad
◆◆◆
To find the arccot, arcsec, and arccsc, we first take the reciprocal of the given function and
then find the inverse function, as shown in the following example.
◆◆◆
Remember that the inverse
trigonometric function can be
written in two different ways. Thus
the inverse sine can be written
arcsin or
1
sin
Example 10: If cot1 2.745, find in radians to four decimal places.
Solution: If the cotangent of is 2.745, then
cot 2.745
so
1
tan Also recall that there are infinitely
many angles that have a
particular value of a trigonometric
function. Of these,we are finding
just the smallest positive angle.
2.745
Taking reciprocals of both sides gives
1
tan 0.3643
2.745
Section 16–1
◆
429
Radian Measure
So
tan1 0.3643 0.3494
◆◆◆
To find the trigonometric function of an angle in radians expressed in terms of using the
calculator, it is necessary first to convert the angle to decimal form.
◆◆◆
Example 11: Find cos(5/12) to four significant digits.
Solution: Converting 5/12 to decimal form, we have
5()/12 1.3090
With our calculator in radian mode, we then take the cosine.
cos 1.3090 rad 0.2588
◆◆◆
◆◆◆
Example 12: Evaluate to four significant digits:
5 cos(2/5) 4 sin2(3/7)
Solution: From the calculator,
2
cos 0.309 02 and
5
3
sin 0.974 93
7
so
3
2
5 cos 4 sin2 5(0.309 02) 4(0.974 93)2
7 5.347
5
◆◆◆
Areas of Sectors and Segments
Sectors and segments of a circle (Fig. 16–3) were defined in Sec. 6–4. Now we will
compute their areas.
The area of a circle of radius r is given by r2, so the area of a semicircle, of course,
is r2/2; the area of a quarter circle is r2/4; and so on. The sector area is the same fractional part of the whole area as the central angle is of the whole.
Thus if the central angle is 1/4 revolution, the sector area is also 1/4 of the total
circle area.
If the central angle (in radians) is /2 revolution, the sector area is also /2 the
total circle area. So
area of sector r2 p q
2
Area of
a Sector
◆◆◆
r2
Area 2
where is in radians
h
r
Sector
FIGURE 16–3 Segment and sector
of a circle.
116
Example 13: Find the area of a sector having a radius of 8.25 m and a central angle of 46.8.
Solution: We first convert the central angle to radians.
46.8( rad/180) 0.8168 rad
Then, by Eq. 116,
(8.25)2 (0.8168)
area 27.8 m2
2
Segment
◆◆◆
430
Chapter 16
◆
Radian Measure, Arc Length, and Circular Motion
The area of a segment of a circle (Fig. 16–3) is
rh
Area r2 arccos (r h) 兹2rh h2
r
Area of a
Segment
rh
where arccos is in radians
r
An alternate equation for calculating the area of a segment is:
r2( sin )
Area 2
Area of a
Segment
(alternate
equation)
[
]
(r h)
where 2 arccos (in radians)
r
◆◆◆
Example 14: Compute the area of a segment having a height of 10.0 cm in a circle of
radius 25.0 cm.
Solution: Substituting into the given equation gives
25.0 10.0
Area (25.0)2 arccos (25.0 10.0) 兹 2(25.0)(10.0) (10.0)2
25.0
625 arccos 0.600 15.0 兹500 _ 100
625(0.9273) 15.0(20.0) 280 cm2
The same solution can be found using the alternative equation for the area of a segment:
First, find 2 arccos [(25.0 10.0)/25.0]
106.3
106.3 (/180) rad
1.855 rad
Then, substituting into the equation r2( sin )/2, we obtain:
2
2
Area 25.0
_ (1.855 sin 106.5)/2 279.7 cm
280 cm2
Exercise 1
◆
Radian Measure
Convert to radians.
1. 47.8
4. 0.370 rev
2. 18.7
5. 1.55 rev
3. 3515
6. 1.27 rev
8. 2.30 rad
11. 1.12 rad
9. 3.12 rad
12. 0.766 rad
14. 4.275 rad
17. 1.14 rad
15. 0.372 rad
18. 0.116 rad
Convert to revolutions.
7. 1.75 rad
10. 0.0633 rad
Convert to degrees (decimal).
13. 2.83 rad
16. 0.236 rad
◆◆◆
Section 16–1
◆
431
Radian Measure
Convert each angle given in degrees to radian measure in terms of .
19. 60
20. 130
21. 66
22. 240
23. 126
24. 105
25. 78
26. 305
27. 400
28. 150
29. 81
30. 189
Convert each angle given in radian measure to degrees.
2
31. 32. 8
3
3
34. 35. 5
9
5
7
38. 37. 8
9
6
40. 41. 7
12
9
33. 11
4
36. 5
2
39. 15
8
42. 9
Calculator
Use a calculator to evaluate to four significant digits.
44. tan 0.442
43. sin 3
3
2
46. tan p q
47. cos 3
5
4
49. sec 0.355
50. csc 3
9
6
p
52. tan 53. cos q
11
5
55. cos 1.832
56. cot 2.846
58. csc 0.8163
59. arcsin 0.7263
62. arccot 1.546
61. cos1 0.2320
1
65. arctan 3.7253
64. csc 2.6263
2 68. 7 tan2 67. sin cos 9
6
6
70. sin 71. sin tan 6
8
8
6
45. cos 1.063
7
48. sin p q
8
8
51. cot 9
54. sin 1.075
57.
60.
63.
66.
sin 0.6254
arccos 0.6243
sin1 0.2649
arcsec 2.8463
3
69. cos2 4
72. 3 sin cos2 9
9
73.
74.
75.
76.
77.
Find the area of a sector having a radius of 5.92 cm and a central angle of 62.5.
Find the area of a sector having a radius of 3.15 m and a central angle of 28.3.
Find the area of a segment of height 12.4 cm in a circle of radius 38.4 cm.
Find the area of a segment of height 55.4 cm in a circle of radius 122.6 cm.
A weight bouncing on the end of a spring moves with simple harmonic motion according
to the equation y 4 cos 25t, where y is in inches. Find the displacement y when t 2.00 s.
(In this equation, the angle 25t must be in radians.)
78. The angle D (measured at the earth’s centre) between two points on the earth’s surface is
found by
cos D sin L1 sin L2 cos L1 cos L2 cos(M1 M2)
where L1 and M1 are the latitude and longitude, respectively, of one point, and L2 and M2
are the latitude and longitude of the second point. Find the angle between Yarmouth, NS
(latitude 44 N, longitude 66 W), and Whitehorse, YT (61 N, 135 W), by substituting
into this equation.
432
Chapter 16
◆
Radian Measure, Arc Length, and Circular Motion
79. A grinding machine for granite uses a grinding disk that has four abrasive pads with the
dimensions shown in Fig. 16–4. Find the area of each pad.
80. A partial pulley is in the form of a sector with a cylindrical hub, as shown in Fig. 16–5.
Using the given dimensions, find the volume of the pulley.
Disk,
62.8 cm
radius
Hole,
8.50 cm
radius
5.270 in. radius
Hole dia.
0.500 in.
67.5˚
55.8˚
Abrasive
pad
FIGURE 16–4
16–2
Hub dia.
1.250 in.
0.250 in.
FIGURE 16–5
Arc Length
We have seen that arc length, radius, and central angle are related to each other. In
Fig. 16–6, if the arc length s is equal to the radius r, we have equal to 1 radian. For
other lengths s, the angle is equal to the ratio of s to r.
r
s
Central
Angle
s
r
where is in radians
115
The central angle in a circle (in radians) is equal to the ratio of the intercepted arc and
the radius of the circle.
FIGURE 16–6 Relationship
between arc length, radius,
and central angle.
When you are dividing s by r to obtain , s and r must have the same units. The units cancel, leaving as a dimensionless ratio of two lengths. Thus the radian is not a unit of measure
like the degree or inch, although we usually carry it along as if it were.
We can use Eq. 115 to find any of the quantities , r, or s when the other two are known.
◆◆◆
Example 15: Find the angle that would intercept an arc of 27.0 m in a circle of radius 21.0 m.
Solution: From Eq. 115,
27.0 m
s
1.29 rad
r
21.0 m
◆◆◆
◆◆◆
Example 16: Find the arc length intercepted by a central angle of 62.5 in a 10.4-cm-radius
circle.
Solution: Converting the angle to radians, we get
rad
62.5 p q 1.09 rad
180
By Eq. 115,
s r 10.4 cm(1.09) 11.3 cm
◆◆◆