Spherical Trigonometry and non Euclidean Geometry

OK, this might be a little specialist, but I have confidence in JREF's unbounded talents :)



I'm reading a fascinating book on non-Euclidean geometry - one of the sections looks at how Taurinus created a non-Euclidean geometry by having the normal spherical formula for trigonometry on a sphere with imaginary radius...and this gives the formula below:



Cosh(a/k) = cosh(b/k).cosh(c/k) - sinh(b/k).sinh(c/k)cosA



Now, the book then states that when we use the exponential forms of the hyperbolic functions, we see that as k tends to infinity we are left with the familiar:



a^2 = b^2 +c^2 - 2bc CosA



Now that's pretty incredible - as it means that our "normal" Euclidean geometry would be a special case of a wider generalised geometry. However, when I try and let k tend to infinity I don't get anything like the required formula (I actually get 1 = 1 which whilst true isn't very interesting!)



I simply substituted the exponential forms to give:



0.5(e^(a/k) + e^(-a/k) )= RHS etc etc



but then when k tends to infinity for each exponential term we have e^(a/k) and e^(-a/k) both tending towards 1. Which ultimately gives 1 = 1....



Any ideas?



Thanks :)

via JREF Forum http://ift.tt/1mf25Uo