599 6.1 Radian Measure 39.728 0.8725 ft Figure 8 EXAMPLE 5 Finding a Length Using s =r U A rope is being wound around a drum with radius 0.8725 ft. (See Figure 8.) How much rope will be wound around the drum if the drum is rotated through an angle of 39.72°? SOLUTION The length of rope wound around the drum is the arc length for a circle of radius 0.8725 ft and a central angle of 39.72°. Use the formula s = r u, with the angle converted to radian measure. The length of the rope wound around the drum is approximated by s. s = r u = 0.8725c 39.72 a p 180b d ≈0.6049 ft S Now Try Exercise 87(a). Convert to radian measure. EXAMPLE 6 Finding an Angle Measure Using s =r U Two gears are adjusted so that the smaller gear drives the larger one, as shown in Figure 9. If the smaller gear rotates through an angle of 225°, through how many degrees will the larger gear rotate? SOLUTION First find the radian measure of the angle of rotation for the smaller gear, and then find the arc length on the smaller gear. This arc length will correspond to the arc length of the motion of the larger gear. Because 225° = 5p 4 radians, the smaller gear has arc length as follows. s = r u = 2.5 a 5p 4 b = 12.5p 4 = 25p 8 cm The tips of the two mating gear teeth must move at the same linear speed, or the teeth will break. So we must have “equal arc lengths in equal times.” An arc with this length s on the larger gear corresponds to an angle measure u, in radians, where s = r u. s = r u Arc length formula 25p 8 = 4.8u Let s = 25p 8 and r = 4.8 (for the larger gear). 125p 192 = u 4.8 = 48 10 = 24 5 ; Multiply by 5 24 to solve for u. Converting u back to degrees shows that the larger gear rotates through 125p 192 a 180° p b ≈117°. Convert u = 125p 192 to degrees. S Now Try Exercise 81. 2.5 cm 4.8 cm Figure 9 r The shaded region is a sector of the circle. U Figure 10 Area of a Sector of a Circle A sector of a circle is the portion of the interior of a circle intercepted by a central angle. Think of it as a “piece of pie.” See Figure 10. A complete circle can be thought of as an angle with measure 2p radians. If a central angle for a sector has measure u radians, then the sector makes up the fraction u 2p of a complete circle. The area of a complete circle with radius r is = pr2. Therefore, we have the following. Area of a sector = u 2p 1pr22 = 1 2 r2 u, where u is in radians.
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