598 CHAPTER 8 Applications of Trigonometric Functions 5. Replacing the ratios in part (4) by the cosines of the sides of the spherical triangle, you should now have the Law of Cosines for spherical triangles: = + c a b a b C cos cos cos sin sin cos Source: For the spherical Law of Cosines, see Mathematics from the Birth of Numbers by Jan Gullberg. W. W. Norton & Co., Publishers, 1996, pp. 491–494. II. The Lewis and Clark Expedition Lewis and Clark followed several rivers in their trek from what is now Great Falls, Montana, to the Pacific coast. First, they went down the Missouri and Jefferson rivers from Great Falls to Lemhi, Idaho. Because the two cities are at different longitudes and different latitudes, we must account for the curvature of Earth when computing the distance that they traveled. Assume that the radius of Earth is 3960 miles. 1. Great Falls is at approximately ° 47.5N and ° 111.3 W. Lemhi is at approximately ° 45.5N and ° 113.5 W. (We will assume that the rivers flow straight from Great Falls to Lemhi on the surface of Earth.) This line is called a geodesic line. Use the Law of Cosines for a spherical triangle [see Project I, part (5)] to find the angle between Great Falls and Lemhi. (The central angles are found by using the differences in the latitudes and longitudes of the towns. See the figure.) Then find the length of the arc joining the two towns. (Recall θ = s r .) Lemhi Great Falls North South a b c 2. From Lemhi, they went up the Bitteroot River and the Snake River to what is now Lewiston and Clarkston on the border of Idaho and Washington. Although this is not really a side to a triangle, we will make a side that goes from Lemhi to Lewiston and Clarkston. If Lewiston and Clarkston are at about ° 46.5 N 117.0 W, ° find the distance from Lemhi using the Law of Cosines for a spherical triangle and the arc length. 3. How far did the explorers travel just to get that far? 4. Draw a plane triangle connecting the three towns. If the distance from Lewiston to Great Falls is 282 miles and the angle at Great Falls is ° 42 and the angle at Lewiston is ° 48.5 , find the distance from Great Falls to Lemhi and from Lemhi to Lewiston. How do these distances compare with the ones computed in parts (1) and (2)? Source: For Lewis and Clark Expedition: American Journey: The Quest for Liberty to 1877, Texas Edition. Prentice Hall, 1992, p. 345. Source: For map coordinates: National Geographic Atlas of the World, published by National Geographic Society, 1981, pp. 74–75. Citation: Used with permission of Technology Review, from W. Roush, “From Lewis and Clark to Landsat: David Rumsey’s Digital Maps Marry Past and Present,” 108, no. 7, © 2005; permission conveyed through Copyright Clearance Center, Inc.
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