1) Given that a = 1, b= 2, k = 1 and we need to find c.

We have: cosh(c/k) = cosh (a/k) cosh(b/k)

Or: Letting k = 1:

cosh(c) = cosh (a)
cosh(b)

cosh(c) = cosh(1) cosh(2) = (1.543)(3.762) = 5.804

Take the inverse hyperbolic cosine:

cosh

^{-1}(c) = cosh^{-1}(5.804) = c
Or:

c = 2.444

The ratio of the sines of the non-Euclidean angles A, B is
given by:

sin (A)/ sin(B) =
{(sinh(a/k)/ sinh(b/k)}

so that, since k = 1:

sin (A)/ sin(B) = {(sinh(a) / sinh(b)}

sin (A)/ sin(B) = {sinh
(1)/ sinh (2)}

where: sinh(1) = 1.175 and sinh(2) = 3.626

Therefore:

sin (A)/ sin(B) = (1.175)/ (3.626) = 0.323

2) We compute the ratio x/k
= 2 (i.e. 1.0 / 0.5 = 2.0; 0.5/ 0.25 =
2.0 etc.)

Then: exp (- x/ k)
= exp (-2) = 0.1353=
1 / (exp(2))

But the curvature:

**k**= -1/ k^{2}^{ }

To conform with: 1 /
(exp(x/k)) = 1 (exp (2)) we must have k = 1 (positive real no.)

Therefore:

**k**= -1/ k^{2 }= -1 / (1)^{ 2}= -1
Hence, the curvature is negative, and the space is

*hyperbolic.*^{}
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