Here we prove results about relations between the angles and
the hyperbolic lengths of the sides of hyperbolic triangles.
Many are analogues of euclidean theorems, but involve various
hyperbolic functions of the lengths, but we must expect an
additional result reflecting the (AAA) condition for h-congruence.
As in euclidean geometry, the results we obtain allow us to
The traditional approach to trigonometry begins with theorems about
The Cosine Rule for Hyperbolic Triangles
Suppose that we know the values of c, b and A, i.e.the lengths
Likewise, if we know a, b and c, then we can determine the angles.
We shall also apply our results to right-angled h-triangles. Our first
Pythagoras's Theorem for Hyperbolic Triangles
We then deduce
The Cosine Formula for Hyperbolic Triangles
Again from the Cosine Rule, we derive
The Sine Rule for Hyperbolic Triangles
Noting that sin(A) = 1 if A = π/2, we have
The Sine Formula for Hyperbolic Triangles
The Sine and Cosine Formulae allow us to derive
The Tangent Formula for Hyperbolic Triangles
In hyperbolic geometry, we have the (AAA) condition for h-congruence.
This implies that, if the corresponding angles of two h-triangles are equal,
then the corresponding sides are equal. In other words, the angles must
determine the lengths of the sides. We can show this algebraically.
The Second Cosine Rule for Hyperbolic Triangles