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 hcongruence.
As in euclidean geometry, the results we obtain allow us to
Standard notation
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 rightangled htriangles. Our first
Pythagoras's Theorem for Hyperbolic Triangles
Proof 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 hcongruence. This implies that, if the corresponding angles of two htriangles 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 proof of the second cosine rule

