486 CHAPTER 7 Analytic Trigonometry Problems 89 and 90 require the following discussion: The shortest distance between two points on Earth’s surface can be determined from the latitude and longitude of the two locations. For example, if location 1 has α β ( ) ( ) = lat, lon , 1 1 and location 2 has α β ( ) ( ) = lat, lon , , 2 2 the shortest distance between the two locations is approximately α β α β α β α β α α ( ) ( ) ( ) [ ] = + + − d rcos cos cos cos cos cos sin cos sin sin sin 1 1 1 2 2 1 1 2 2 1 2 where = ≈ r radius of Earth 3960 miles and the inverse cosine function is expressed in radians. Also, N latitude and E longitude are positive angles, and S latitude and W longitude are negative angles. City Latitude Longitude Chicago, IL ° ′ 41 50N ° ′ 87 37W Honolulu, HI ° ′ 21 18N ° ′ 157 50W Melbourne, Australia ° ′ 37 47S ° ′ 144 58E Source: www.infoplease.com 89. Shortest Distance from Chicago to Honolulu Find the shortest distance from Chicago, latitude ° ′ 41 50 N, longitude ° ′ 87 37W, to Honolulu, latitude ° ′ 21 18N, longitude ° ′ 157 50 W. Round your answer to the nearest mile. 90. Shortest Distance from Honolulu to Melbourne, Australia Find the shortest distance from Honolulu to Melbourne, Australia, latitude ° ′ 37 47 S, longitude ° ′ 144 58 E. Round your answer to the nearest mile. 91. Challenge Problem Solve: ( ) ( ) = − − x cos sin tan cos 4 5 1 1 92. Challenge Problem Find u in terms of x and r: ( ) ( ) = − − x r u tan cos sin tan 1 1 ‘Are You Prepared?’ Answers 1. Domain: the set of all real numbers; Range: − ≤ ≤ y 1 1 2. [ )∞ 3, 3. True 4. 1; − − 3 2 ; 1 2 ; 1 Problems 93–102 are based on previously learned material. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for subsequent sections, a final exam, or later courses such as calculus. Retain Your Knowledge 93. Solve: − + ≤ x3 2 5 9 94. State why the graph of the function f shown is one-to-one. Then draw the graph of the inverse function −f .1 [Hint: The graph of = y x is given.] 95. The exponential function ( ) = + f x 1 2x is one-to-one. Find −f .1 96. Factor: ( ) ( ) ( ) ( ) + + − + + − − − x x x x x 2 1 3 3 21 1 2 2 1 2 2 3 2 1 2 97. Solve: + = e 7 10 x4 98. The diameter of each wheel of a bicycle is 20 inches. If the wheels are turning at 336 revolutions per minute, how fast is the bicycle moving? Express the answer in miles per hour, rounded to the nearest integer. 99. Find the exact value of π π sin 3 cos 3 . 100. If θ = cos 24 25 , find the exact value of each of the remaining five trigonometric functions of acute angle θ. 101. If sin θ > 0 and cot θ < 0, name the quadrant in which the angle θ lies. 102. Find the average rate of change of ( ) = f x x tan from π 6 to π 4 . x y 4 24 4 24 y = x (2, 1) (1, 0) ( , 21) 1 – 2
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