Math 2412 Chapter 6 Notes Spring 2017.pdf

Math 2412 Chapter 6 Notes Spring 2017.pdf - 6.1 Radian...

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128 6.1 Radian Measure Radian An angle with its vertex at the center of a circle that intercepts an arc on the circle equal in length to the radius of the circle has a measure of 1 radian. Unlike degrees, radians can be used to compute distance. y x Converting Between Degrees and Radians 1. Multiply a degree measure by 180 radians to convert to radians. 2. Multiply a radian measure by o 180 to convert to degrees. Caution: When no unit of angle measure is given, it is understood that the angle is measured in radians.
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129 Convert each degree measure to radians. Leave answers as multiples of π. 565.8 30 o Solution radians o 180 30 30 rad rad 6 180 30 # 565.14 -315 o Solution LTR
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130 565.44 264.9 o Solution LTR 565.46 174 o 50’ Solution LTR
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131 Convert each radian measure to degrees. 565.26 3 8 Solution o 180 3 8 3 8 o o 480 3 1440 # 565.32 5 8 Solution LTR
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132 Convert each radian measure to degrees. Write answers to the nearest minute. 565.54 5 Solution LTR 565.58 9.84763 Solution LTR
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133 Arc Length The length s of the arc intercepted on a circle of radius r by a central angle of measure θ radians is given by s = rθ, where θ is measured in radians. Find the length to three significant digits of each arc intercepted by a central angle θ in a circle of radius r. 567.74 r = 0.892 cm, 10 11 rad Solution r s cm cm 08 . 3 10 11 892 . 0 # 567.78 r = 71.9 cm, θ = 135 o Solution LTR
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134 Find the distance in kilometers between each pair of cities, assuming they lie on the same north-south line. 567.84 Farmersville, California, 36 o N, and Penticton, British Columbia, 49 o N. Solution LTR
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135 567.89 Two gears are adjusted so that the smaller gear drives the larger one. If the smaller gear rotates through an angle of 300 o , through how many degrees will the larger gear rotate? Solution First, convert 300 o to radians. radians o 180 300 300 rad 3 5 . Now, we find the arc length of the smaller gear. r s . 30 185 3 5 . 18 3 5 7 . 3 cm cm cm An arc with this length on the la rger gear corresponds to an angle θ, where s = rθ. Now, we find θ for the larger gear. 1 . 7 30 185 r s 213 185 . Now, we convert θ to degrees.
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136 o o 71 11100 180 213 185 . 156 o # Area of a Sector The area A of a sector of a circle of radius r and central angle θ is given by 2 2 1 r A , θ in radians. Find the area of a sector of a circle having radius r and central angle θ. Express answers to the nearest tenth. 569.16 r = 90.0 km, θ = 270 o Solution First, we convert 270 o to radians. radians o 180 270 270 rad 2 3 .
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