I attached the problems. I got problem 1, so just do problem 2,3,4. Thank you so much!

 

 

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. 3. Consider the triangle 4ABC Where A~=   0 0 1  , B~=   1 0 0   and C~=   0 √ 1 2 √ 1 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 = π 2 , ∠B = π 3 , ∠C = π 4 ). (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