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CHAPTER 1 Angles and Applications
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Find the radius of the circle for which an arc 15 in long subtends an angle of (a) 1 rad, (b) (d) 20 , (e) 50 . Ans. (a) 15 in, (b) 22.5 in, (c) 5 in, (d) 43.0 in, (e) 17.2 in
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The end of a 40-in pendulum describes an arc of 5 in. Through what angle does the pendulum swing Ans.
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A train is traveling at the rate 12 mi/h on a curve of radius 3000 ft. Through what angle has it turned in 1 min Ans. 0.352 rad or 20 10 or 20.17
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A curve on a railroad track consists of two circular arcs that make an S shape. The central angle of one is 20 with radius 2500 ft and the central angle of the other is 25 with radius 3000 ft. Find the total length of the two arcs. Ans. 6250 /9 or 2182 ft
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Find the area of the sector determined by a central angle of /3 rad in a circle of diameter 32 mm. Ans. 128 /3 or 134.04 mm2
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Find the central angle necessary to form a sector of area 14.6 cm2 in a circle of radius 4.85 cm. Ans. 1.24 rad or 71.05 or 71 3
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Find the area of the sector determined by a central angle of 100 in a circle with radius 12 cm. Ans. 40 or 125.7 cm2
If the area of a sector of a circle is 248 m2 and the central angle is 135 , find the diameter of the circle. Ans. diameter 29.0 m
A flywheel of radius 10 cm is turning at the rate 900 r/min. How fast does a point on the rim travel in m/s Ans. 3 or 9.4 m/s
An automobile tire has a diameter of 30 in. How fast (r/min) does the wheel turn on the axle when the automobile maintains a speed of 45 mi/h Ans. 504 r/min
In grinding certain tools the linear velocity of the grinding surface should not exceed 6000 ft/s. Find the maximum number of revolutions per second of (a) a 12-in (diameter) emery wheel and (b) an 8-in wheel. Ans. (a) 6000/ r/s or 1910 r/s, (b) 9000/ r/s or 2865 r/s
If an automobile wheel 78 cm in diameter rotates at 600 r/min, what is the speed of the car in km/h Ans. 88.2 km/h
Trigonometric Functions of a General Angle
2.1 Coordinates on a Line
A directed line is a line on which one direction is taken as positive and the other as negative. The positive direction is indicated by an arrowhead. A number scale is established on a directed line by choosing a point O (see Fig. 2.1) called the origin and a unit of measure OA 1. On this scale, B is 4 units to the right of O (that is, in the positive direction from O) and C is 2 units to the left of O (that is, in the negative direction from O). The directed distance OB 4 and the directed distance OC 2. It is important to note that since the line is directed, OB BO and OC CO. The directed distance BO 4, being measured contrary to the indicated positive direction, and the directed distance CO 2. Then CB CO OB 2 4 6 and BC BO OC 4 ( 2) 6.
Fig. 2.1
2.2 Coordinates in a Plane
A rectangular coordinate system in a plane consists of two number scales (called axes), one horizontal and the other vertical, whose point of intersection (origin) is the origin on each scale. It is customary to choose the positive direction on each axis as indicated in the figure, that is, positive to the right on the horizontal axis or x axis and positive upward on the vertical or y axis. For convenience, we will assume the same unit of measure on each axis. By means of such a system the position of any point P in the plane is given by its (directed) distances, called coordinates, from the axes. The x-coordinate of a point P (see Fig. 2.2) is the directed distance BP OA and the
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