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26 Jan 2017: Conversely, show that for any three positive numbers a,b,c less than πsatisfying the above conditions, we have cos(b c) < cos a < cos(b c), and that there isa ... Deduce the formula. sin α sin β cos c = cos γ cos α cos β.

13 Dec 2017: 5. Use the contraction mapping theorem to show that the equation x = cos x has a uniquereal solution. ... sin y1y′1 cos y2y. ′2 = e. y′1. cos y2y′1 sin y1y.

2 Mar 2017: S2 be given byσ(u,v) = (sin u cos v, sin u sin v, cos u). ... 14. Show that Mercator’s parametrization of the sphere (minus poles). σ(u,v) = (sech u cos v, sech u sin v, tanh u).

14 Feb 2017: 16. Let 4 be a hyperbolic triangle, with angles α, β, γ, and sides of length a, b, c (the sideof length a being opposite the vertex with angle α, and ... cosh c = cosh a cosh b sinh a sinh b cos γ.[For a slicker proof of this result, and of the