What are trig classes like within a universe that's "noticeably" hyperbolic? [I want to think that this question has an answer, but it may be more a "community wiki" discussion. Feel free to re-tag.]

What are trig classes like within a universe that's "noticeably"[*] hyperbolic?

Using an appropriate model to peek into such a universe from our Euclidean one shows us that all the fundamental trig relations (Law of Sines, Law of Cosines, etc) involve transcendental functions in both angle measures and lengths. But, then, in our earliest mathematical days, similar triangles allowed us to build a theory of (circular) trig functions based on ratios of lengths of sides, and eventually our mathematics became sophisticated enough to include hyperbolic functions and to be comfortable enough with them and how they might apply to our non-Euclidean models. (We also had the benefit of being able to interpret products as areas of rectangles whose side-lengths correspond to factors.) What if you don't --can't-- draw from our experience base?
How do you even get started developing trigonometry (or explaining it to your hyperbolic trig students[**]) without a concept of similar triangles?
Obviously(?), the Unit Circle is out.
It makes some sense that the Angle of Parallelism would be the fundamental bridge between angle-information and distance-information; one can imagine that hyperbolic people would be "aware" of the phenomenon on some level, and we know that it's a "universal" property. Even so, the most-concise representations of the AoP relationship are transcendental in both angle measure and length. How insightful would a hyperbolic mathematician have to be to discern the equations from tables of observed measurements? And is there a clear path from those equations to, say, (what we know as) the Law of Sines and the Laws of Cosines?
Or perhaps the fundamental figure in an "intrinsic" hyperbolic trig class is the Equilateral Triangle. This idea actually exploits non-similarity to set up a bridge between angle-information and distance-information (with area-information thrown in as a bonus); and it seems that it might be more likely to provide a path to the Laws of Sines and Cosines, since it already relates angles and sides of triangles. But, is it really a particularly helpful starting point? Can you get from there to the Angle of Parallelism relation?
Something else?
[*] For instance, anyone can pull out a protractor and easily see that triangles have an angle-sum less than 180 degrees.
[**] "hyperbolic" modifying both "trig" and "students". :)
 A: Chapter V of Harold E. Wolfe's book:  Introduction to Non-Euclidean Geometry (Holt, Rinehart, and Winston), 1945 (and reprinted, 1966) is entitled: Hyperbolic Plane Trigonometry, and has a systematic treatment of this topic.
A: One model for the hyperbolic plane is Minkowski space.  This is 2-space, say, with the caveat that the dot product is negative on the last coordinate, i.e. $(x_1,y_1)\cdot(x_2,y_2)=x_1x_2 -y_1y_2$.  If we want to define the length of a vector to be the square root of a vector dotted with itself in this inner product, we need to require that $x^2\ge y^2$.  One can then check that the point $(0,1)$ is distance $1$ from the curve $(t,1-\sqrt{t^2-1})$, and likewise for generic points so that the distribution of points equidistant from a fixed point is a hyperbola.  Using calculus, one can compute the length of the arc along the "unit" hyperbola starting at $(0,1)$.  This is how the Greeks defined angles, if I am not mistaken, though they did so without calculus.  One can then define the hyperbolic cosine as the inverse function of the length of the above curve, and so forth.
I think this is probably how it would begin.  Defining the hyperbolic trig functions as combinations of exponentials would likely come later.  Perhaps there is a way to come up with the length of the hyperbolic arc without calculus?
A: I'm more comfortable with how trigonometry and hyperbolic geometry actually
did happen than how they might happen in a hypothetical hyperbolic world.
So, this may not be what you're looking for, but I hope it sheds some light.
First, similar triangles did not lead immediately to trigonometry. There are
similar triangles in Euclid, but not trigonometry. In fact, the first 
trigonometry to develop was spherical trigonometry, for use in astronomy.
Second, hyperbolic functions were introduced by Lambert (around 1760), 
who observed that they were the trigonometric functions on a "sphere of 
imaginary radius" -- which we now realize can be viewed as a hyperbolic plane.
Third, around 1830 hyperbolic functions were found by Lobachevsky to describe 
the trigonometry of the non-Euclidean plane that he was studying axiomatically.
He even attempted to claim that formulas he found were a "realization" of the
non-Euclidean plane. About the same time, in ignorance of Lobachevsky's work,
Minding found that hyperbolic functions describe the trigonometry of any
surface of constant negative curvature.
Finally, Beltrami in 1868 showed that in hyperbolic space there are surfaces
that realize both spherical and Euclidean geometry. So, even in a hyperbolic
world, one has the option of using spherical or Euclidean geometry --
which are easier. So creatures in the hyperbolic world might still follow
the path that we did. Students in trig class might use "horosphere paper"
so as to take advantage of Euclidean trig.
