572 CHAPTER 5 Trigonometric Functions Use the formula relating distance, rate, and time to find the distances in Figure 54 from A to C and from C to B. b = 22 * 3.5 = 77 nautical mi Distance = rate * time a = 22 * 4.0 = 88 nautical mi Now find c, the distance from port at point A to the ship at point B. a2 + b2 = c2 Pythagorean theorem 882 + 772 = c2 a = 88, b = 77 c = 2882 + 772 If a2 + b2 = c2 and c 70, then c = 2a2 + b2. c ≈120 nautical mi Two significant digits S Now Try Exercise 75. B D N A Port 438 478 S S C 478 N c b = 77 nautical mi a = 88 nautical mi Figure 54 (repeated) Further Applications EXAMPLE 7 UsingTrigonometry to Measure a Distance The subtense bar method is a method that surveyors use to determine a small distance d between two points P and Q. The subtense bar with length b is centered at Q and situated perpendicular to the line of sight between P and Q. See Figure 55. Angle u is measured, and then the distance d can be determined. b/2 b/2 P d Q u Figure 55 (a) Find d when u = 1° 23′ 12″ and b = 2.0000 cm. (b) How much change would there be in the value of d if u measured 1″ larger? SOLUTION (a) From Figure 55, we obtain the following. cot u 2 = d b 2 Cotangent ratio d = b 2 cot u 2 Multiply and rewrite. Let b = 2 . To evaluate u 2 , we change u to decimal degrees. 1° 23′ 12″ ≈1.386666667° Then d = 2 2 cot 1.386666667° 2 ≈82.634110 cm. (b) If u is 1″ larger, then u = 1° 23′ 13″ ≈1.386944444°. d = 2 2 cot 1.386944444° 2 ≈82.617558 cm The difference is 82.634110 - 82.617558 = 0.016552 cm. S Now Try Exercise 87. Use cot u = 1 tan u to evaluate.
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