Pre-Calc Exam Notes 50

Pre-Calc Exam Notes 50 - center of earth r e earth station...

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50 Chapter 2 General Triangles §2.2 17. The Dutch astronomer and mathematician Willebrord Snell (1580-1626) wrote the Law of Cosines as 2 ab c 2 ( a b ) 2 = 1 1 cos C in his trigonometry text Doctrina triangulorum (published a year after his death). Show that this formula is equivalent to formula (2.11) in our statement of the Law of Cosines. 18. Suppose that a satellite in space, an earth station, and the center of the earth all lie in the same plane. Let r e be the radius of the earth, let r s be the distance from the center of the earth to the satellite (called the orbital radius of the satellite), and let d be the distance from the earth station to the satellite. Let E be the angle of elevation from the earth station to the satellite, and let γ and ψ be the angles shown in Figure 2.2.5.
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Unformatted text preview: center of earth r e earth station d r s local horizontal E satellite Figure 2.2.5 Use the Law of Cosines to show that d = r s r 1 + p r e r s P 2 2 p r e r s P cos , and then use E = 90 and the Law of Sines to show that cos E = sin r 1 + p r e r s P 2 2 p r e r s P cos . Note: This formula allows the angle of elevation E to be calculated from the coordinates of the earth station and the subsatellite point (where the line from the satellite to the center of the earth crosses the surface of the earth). 3 3 See pp. 22-25 in T. PRATT AND C.W. BOSTIAN, Satellite Communications , New York: John Wiley & Sons, 1986....
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