806 CHAPTER 8 Applications of Trigonometry 425 1248 36.0 c b a N S E W N1 S1 E1 W1 a A B F C 239° 59° 25° 115° NOT TO SCALE Figure 32 Vector c represents the airspeed and bearing of the plane, vector a represents the speed and direction of the wind, and vector b represents the resulting bearing and ground speed of the plane. Angle ABC has as its measure the sum of angle ABN1 and angle N1BC. • Angle SAB measures 239° - 180° = 59°. Because angle ABN1 is an alternate interior angle to it, ABN1 = 59°. • Angle E1BF measures 115° - 90° = 25°. Thus, angle CBW1 also measures 25° because it is a vertical angle. Angle N1BC is the complement of 25°, which is 90° - 25° = 65°. By these results, angle ABC = 59° + 65° = 124°. To find b , we use the law of cosines. b 2 = a 2 + c 2 - 2 a c cos ABC Law of cosines b 2 = 36.02 + 4252 - 2136.0214252 cos 124° Substitute. b 2 ≈199,032 Use a calculator. b ≈446 Square root property The ground speed is approximately 446 mph. To find the resulting bearing of b, we must find the measure of angle a in Figure 32 and then add it to 239°. To find a, we use the law of sines. sin a 36.0 = sin 124° 446 sin a = 36.0 sin 124° 446 Multiply by 36.0. a = sin-1 a 36.0 sin 124° 446 b Use the inverse sine function. a ≈4° Use a calculator. Add 4° to 239° to find the resulting bearing of 243°. To maintain accuracy, use all the significant digits that a calculator allows. S Now Try Exercise 51.
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