College Advanced Geometry Problem set. (only do problem 2,3,4)

yvanyvna1995
timer Asked: Mar 31st, 2017

Question Description

I got problem 1, so I need help with problem 2,3,and 4.

The problems are in the attachment

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Math 551 - Spring 2017 - Problem Set 4 Due Friday, March 31 1. Solve the spherical triangle 4ABC if ∠a = 60◦ , ∠b = 70◦ ∠c = 80◦ . (Note: to solve a spherical triangle means to find all 6 of the angles of the spherical triangle ∠A, ∠B, ∠C, ∠a, ∠b, ∠c.) 2. Assume that the earth is a perfect sphere of radius 3960 miles. By the heading, h(~a, P ), of a directed, great circle arc ~a at the point P on the arc we mean the angle made with the “latitude” circle. You are to “file a flight plan” for a direct flight from Syracuse to Cape Town, South Africa. Assume that the map coordinates of Syracuse are 43◦ N , 76◦ W and the map coordinates of Cape Town are 34◦ S, 18◦ E. The flight plan should include the following items: (a) the conversion of the navigational coordinates of Syracuse and Cape Town to spherical coordinates - (φB , θB ) for Syracuse and (φC , θC ) for Cape Town; (b) a complete solution of the triangle with vertices: A (North Pole), B (Syracuse) and C Cape Town; (c) the direction of flight upon leaving Syracuse; (d) the length of your flight in miles and its duration in hours, assuming a constant speed of 700 mph; (e) Identify the when (flight time) and where (longitude) you will cross the equator.       0 0 1 ~  0 , B= ~  0  and C= ~  √1 . 3. Consider the triangle 4ABC Where A= 2 √1 1 0 2 Find the coordinates of the vertices of the dual triangle 4A∗ B ∗ C ∗ . 4. Let 4ABC be a triangle on the unit sphere with ∠A = π2 , ∠B = π3 , ∠C = π4 ). (a) Compute the arc length of a (b) This triangle may be used to tile the sphere; that is, the entire surface of the sphere may be cut up into triangles congruent to 4ABC. How many triangles will there be in such a tiling?
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