Geometry

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Question


1) In the sphere S^2, consider two great semi-circles joining the �North pole� (0,0,1) to the �South Pole� (0,0,-1) and making an angle of ? with each other at these poles. Show that the surface area of the digon bounded by these two arcs is equal to 2?
Hint: Use spherical coordinates or a proportionality argument
2) Let T be a spherical triangle with angles ? ,?, ?. Let A, B, and C be the great circles of S^2 that contain each of the three edges of T . Show that these great circles subdivide the sphere S^2 into eight spherical triangles whose angles are all of the form ? ,?, ? or ? �?, ? – ?, ? �?
3) Combine parts a and b to show that the area of the triangle T is equal to ?+?+? �?.
Hint: Solve a system of linear equations
4) Show that necessarily ? < ?+?+? < ? +2min{?,?,?}.
Hint: First show that 0< ?+?+?-? <2?