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This is the question:

Problem B.1 Two cities - Tel-Aviv, Israel and SanDiego, CA - have the same latitude 32 ◦ N, but, different longitudes: Tel-Aviv is 34 ◦ E and San-Diego is 117 ◦ W. What is the maximal distance between the great circle arc and the 32 ◦ latitude line? By how much the path along the great circle (geodesic) arc will be shorter than the path along the latitude line? Hint: Spherical triangles (on a sphere with radius 1) satisfy a spherical law of cosines $\cos(c)=\cos(a)\cos(b)+\sin(a)\sin(b)\cos(C)$.

I don't want the solution. My main problem is I don't know where to begin and where I want to get to. Could you help me with figuring out how to get through such problems?

Euler-Lagrange equation: $\frac{d}{dx}L_{y'}-L_y=0$ while $I[y(x)]=\int_{x_0}^{x_1}L(y',y,x)dx$

Full description (as our lecturer assures us) is right here, page 2.

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    @copper.hat The geodesic lies on the same plane as earth center, but the latitude slices the sphere parallel to equator. Logically, it is clear that the distance is maximal at $\frac{\alpha}{2}$, but it needs to be proved. If I'm not that clear - you could check out the link in the post. There's the full explanation.2012-04-23

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