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When changing angles in decimals to minutes and seconds, the general rule is that angles in tenths will be changed to the nearest minute and all other angles will be rounded to the nearest hundredth and then changed to the nearest second. When changing angles in minutes and seconds to decimals, the results in minutes are rounded to tenths and angles in seconds have the results rounded to hundredths.
EXAMPLE 1.2 (a) 62.4
62 23
0.4(60 ) 0.9(60 )
62 24 23 54 29 13.8 37 28.2 78.28333. . . 16 /3600 29 13 37 28 0.8(60 ) 0.2(60 )
(b) 23.9 (c) 29.23 (d) 37.47 (e) 78 17
29 0.23(60 ) 29 13 48 37 0.47(60 ) 37 28 12 78 58 17 /60 22 /60
78.3 (rounded to tenths) 58.37111. . . 58.37 (rounded to hundredths)
(f) 58 22 16
A radian (rad) is defined as the measure of the central angle subtended by an arc of a circle equal to the radius of the circle. (See Fig. 1.3.)
Fig. 1.3
The circumference of a circle
2 (radius) and subtends an angle of 360 . Then 2 radians 1 radian 180 p 57.296 57 17r45s
360 ; therefore
1 degree
p radian 180
0.017453 rad
where
CHAPTER 1 Angles and Applications
7 p rad 12 7p # 180 p 12
EXAMPLE 1.3 (a)
105 5p rad 18 30 7p rad 6 (See Probs. 1.1 and 1.2.)
(b) 50 (c) (d)
p 50 # rad 180 p # 180 p 6
p rad 6 210
p 210 # rad 180
1.4 Arc Length
On a circle of radius r, a central angle of radians, Fig. 1.4, intercepts an arc of length s that is, arc length radius central angle in radians. ru
(NOTE: s and r may be measured in any convenient unit of length, but they must be expressed in the same unit.)
Fig. 1.4
EXAMPLE 1.4 (a) On a circle of radius 30 in, the length of the arc intercepted by a central angle of 3 rad is
30 A 3 B
10 in
(b) On the same circle a central angle of 50 intercepts an arc of length s ru 30a 5p b 18 25p in 3
(c) On the same circle an arc of length 11 ft subtends a central angle 2 u u s r s r 18 30 3>2 5>2 3 rad 5 3 rad 5 when s and r are expressed in inches
when s and r are expressed in feet (See Probs. 1.3 1.8.)
CHAPTER 1 Angles and Applications
1.5 Lengths of Arcs on a Unit Circle
The correspondence between points on a real number line and the points on a unit circle, x2 its center at the origin is shown in Fig. 1.5. y2 1, with
Fig. 1.5
The zero (0) on the number line is matched with the point (1, 0) as shown in Fig. 1.5(a). The positive real numbers are wrapped around the circle in a counterclockwise direction, Fig. 1.5(b), and the negative real numbers are wrapped around the circle in a clockwise direction, Fig. 1.5(c). Every point on the unit circle is matched with many real numbers, both positive and negative. The radius of a unit circle has length 1. Therefore, the circumference of the circle, given by 2 r, is 2 . The distance halfway around is and the distance 1/4 the way around is /2. Each positive number is paired with the length of an arc s, and since s r 1. , each real number is paired with an angle in radian measure. Likewise, each negative real number is paired with the negative of the length of an arc and, therefore, with a negative angle in radian measure. Figure 1.6(a) shows points corresponding to positive angles, and Fig. 1.6(b) shows points corresponding to negative angles.
Fig. 1.6
CHAPTER 1 Angles and Applications
1.6 Area of a Sector
The area K of a sector of a circle (such as the shaded part of Fig. 1.7) with radius r and central angle radians is K that is, the area of a sector
1 2 1 2 2r u
the radius
the radius
the central angle in radians.
(NOTE: K will be measured in the square unit of area that corresponds to the length unit used to measure r.)
Fig. 1.7
EXAMPLE 1.5
For a circle of radius 30 in, the area of a sector intercepted by a central angle of 1 rad is 3 K
1 2 1 2 (30) 3
150 in2
EXAMPLE 1.6
For a circle of radius 18 cm, the area of a sector intercepted by a central angle of 50 is K
1 2 2 (18)
5p 18
45p cm2 or 141 cm2 (rounded)
(NOTE:
5 /18 rad.)
(See Probs. 1.9 and 1.10.)
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