Good book on Spherical Trigonometry

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Possible approach/content:

  • Modern
  • Practical (Navigation/Geodesy)
  • unifies with Euclidean/Hyperbolic Trigonometry
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I don't know exactly if this is what you are looking for or if it's at the right level, but you might want to have a look at the first three chapters of John Ratcliffe's Foundations of Hyperbolic Manifolds. They concern Euclidean, spherical and hyperbolic geometry (with sections about trigonometry), and the book is modern and (in my opinion) well-writen.