SECTION 8.2 The Law of Sines 569 63. Challenge Problem Mollweide’s Formula Another form of Mollweide’s Formula is ( ) ( ) − = ⎡ − ⎣ ⎢ ⎤ ⎦ ⎥ a b c A B C sin 1 2 cos 1 2 Derive it. 64. Challenge Problem For any triangle, derive the formula = + a b C c B cos cos 65. Challenge Problem Law of Tangents For any triangle, derive the Law of Tangents: ( ) ( ) − + = ⎡ − ⎣ ⎢ ⎤ ⎦ ⎥ ⎡ + ⎣ ⎢ ⎤ ⎦ ⎥ a b a b A B A B tan 1 2 tan 1 2 66. Challenge Problem Circumscribing a Triangle Show that = = = A a B b C c r sin sin sin 1 2 See the figure, where r is the radius of the circle circumscribing the triangle PQR whose sides are a, b, and c, and ′ = PP r2 is a diameter of the circle. P9 R Q P b c a B C A 60. Venus The distance from the Sun to Earth is approximately 149,600,000 km. The distance from the Sun to Venus is approximately 108,200,000 km. The elongation angle α is the angle formed between the line of sight from Earth to the Sun and the line of sight from Earth to Venus. Suppose that the elongation angle for Venus is ° 10 . Use this information to find the possible distances between Earth and Venus. 61. The Original Ferris Wheel George Washington Gale Ferris, Jr., designed the original Ferris wheel for the 1893 World’s Columbian Exposition in Chicago, Illinois. The wheel had 36 equally spaced cars each the size of a school bus. The distance between adjacent cars was approximately 22 feet. Determine the diameter of the wheel to the nearest foot. Source: Carnegie Library of Pittsburgh, www.clpgh.org 62. Challenge Problem Mollweide’s Formula For any triangle, Mollweide’s Formula (named after Karl Mollweide, 1774–1825) states that ( ) ( ) + = ⎡ − ⎣ ⎢ ⎤ ⎦ ⎥ a b c A B C cos 1 2 sin 1 2 Derive it. [Hint: Use the Law of Sines and then a Sum-to-Product Formula.] Notice that this formula involves all six parts of a triangle. As a result, it is sometimes used to check the solution of a triangle. ‘Are You Prepared?’ Answers 1. − A B A B sin cos cos sin 2. π π { } 6 , 5 6 3. ° = ° = sin40 0.64; sin80 0.98 4. ° 49.5 69. What do you do first if you are asked to solve a triangle and are given two sides and the angle opposite one of them? 70. Solve Example 6 using right triangle geometry. Comment on which solution, using the Law of Sines or using right triangles, you prefer. Give reasons. 67. Make up three problems involving oblique triangles. One should result in one triangle, the second in two triangles, and the third in no triangle. 68. What do you do first if you are asked to solve a triangle and are given one side and two angles? Explaining Concepts Problems 71–80 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 71. Solve: + − − = x x x 3 4 27 36 0 3 2 72. Find the exact distance between ( ) = − − P 1, 7 1 and ( ) = − P 2, 1. 2 Then approximate the distance to two decimal places. 73. Find the exact value of ( ) ⎡ − ⎣ ⎢ ⎤ ⎦ ⎥ − tan cos 7 8 . 1 74. Graph ( ) = y x 4 sin 1 2 . Show at least two periods. 75. Write the equation = a 100 x 0.2 in logarithmic form. 76. Approximate the average rate of change for ( ) = + g x e x 3 ln x2 on the interval [ ] 1, 3 . Round to three decimal places. 77. Find the horizontal asymptote of ( ) = − − + − h x x x x 3 7 1 4 9 2 2 78. Find an equation of the line perpendicular to = − + y x 2 3 7 that contains the point ( ) − − 2, 5. 79. Determine whether ( ) = − + h x x x 5 4 1 3 is even, odd, or neither. 80. Solve: ( ) − + > x x 1 3 6 4 0
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