SECTION 8.1 Right Triangle Trigonometry; Applications 553 In navigation and surveying, the direction or bearing from a point O to a point P equals the acute angle θ between the ray OP and the vertical line through O, the north–south line. Figure 16 illustrates some bearings. Notice that the bearing from O to P1 is denoted by the symbolism N30 E, ° indicating that the bearing is 30° east of north. In writing the bearing from O to P, the direction north or south always appears first, followed by an acute angle, followed by east or west. In Figure 16, the bearing from O to P2 is S50 W, ° and from O to P3 it is N70 W. ° *In air navigation, the term azimuth denotes the positive angle measured clockwise from the north (N) to a ray OP . In Figure 16, the azimuth from O to P1 is 30 ;° the azimuth from O to P2 is 230 ;° the azimuth from O to P3 is 290 .° In naming runways, the units digit is left off the azimuth. Runway 2 LEFT means the left runway with a direction of azimuth 20° (bearing N20 E° ). Runway 23 is the runway with azimuth 230° and bearing S50 W. ° Figure 16 N708W P3 P1 P4 P2 S508W N308E 208 508 708 O 308 N S E W Finding the Bearing of an Object In Figure 16, what is the bearing from O to an object at P ?4 Solution EXAMPLE 11 The acute angle between the ray OP4 and the north–south line through O is 20 .° The bearing from O to P4 is S20 E. ° Solution Figure 17 P Q O 20° 2 u 1 Runway 2 LEFT N S E W EXAMPLE 12 Finding the Bearing of an Airplane A Boeing 777 aircraft takes off from O’Hare Airport on runway 2 LEFT, which has a bearing of N20 E. ° * After flying for 1 mile, the pilot of the aircraft requests permission to turn 90° and head toward the northwest. The request is granted. After the plane goes 2 miles in this direction, what bearing should the control tower use to locate the aircraft? Figure 17 illustrates the situation. After flying 1 mile from the airport O (the control tower), the aircraft is at P. After turning 90° toward the northwest and flying 2 miles, the aircraft is at the point Q. In triangle OPQ , the angle θ obeys the equation tan 2 1 2 so tan 2 63.4 1 θ θ = = = ≈ ° − The acute angle between north and the ray OQ is 63.4 20 43.4 . °− ° = ° The bearing of the aircraft from O to Q is N43.4 W. ° 1. Now Work 1. Modeling 1.ExplainingConcepts Calculus Preview 1.InteractiveFigure ‘Are You Prepared?’ Answers are given at the end of these exercises. If you get a wrong answer, read the pages listed in red. 8.1 Assess Your Understanding 1. In a right triangle, if the length of the hypotenuse is 5 and the length of one of the other sides is 3, what is the length of the third side? (pp. A14–A15) 2. If θ is an acute angle, solve the equation tan 1 2 . θ = Express your answer in degrees, rounded to one decimal place. (pp. 493–496) 3. If θ is an acute angle, solve the equation sin 1 2 . θ = (pp. 493–496) Now Work PROBLEM 6 3
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