trigonometry and hyperbolic triangles

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
determine the values of certain lengths and angles from others.
Once we know enough of the values, all the others can be found.
This amounts to being able to deduce that two triangles are
congruent if they have a sufficient set of equal values. In other
words, they are equivalent to our congruence conditions.

Standard notation
If ABC is an h-triangle, then we write
A for the angle at vertex A, i.e. <BAC, and
a for the hyperbolic length of the side opposite vertex A, i.e. d(B,C).

The traditional approach to trigonometry begins with theorems about
right-angled triangles. These define the trigonometric functions.
Here we begin with the Cosine Rule and deduce the others. This
could also be done in the euclidean case.

The Cosine Rule for Hyperbolic Triangles
For any h-triangle ABC,
sinh(b)sinh(c)cos(A) = cosh(b)cosh(c) - cosh(a),
with similar formulae for cos(B) and cos(C).

proof of the cosine rule

Suppose that we know the values of c, b and A, i.e.the lengths
of two sides and the size of the included angle. Then the Rule
allows us to calculate a. The variants of the Rule then give cos(B)
and cos(C). Since the angles are in the range (0,π), these determine
the angles uniquely. This is equivalent to the (SAS) condition.

Likewise, if we know a, b and c, then we can determine the angles.
This is the (SSS) condition.

We shall also apply our results to right-angled h-triangles. Our first
result is an analogue of the most famous result in geometry.

Pythagoras's Theorem for Hyperbolic Triangles
The h-triangle ABC has a right angle at A
if and only if cosh(a) = cosh(b)cosh(c).

Proof
We need only observe that A =π/2 if and only if cos(A) = 0.

We then deduce

The Cosine Formula for Hyperbolic Triangles
If the h-triangle ABC has a right angle at A, then
cos(B) = tanh(c)/tanh(a), and
cos(C) = tanh(b)/tanh(a).

proof of the cosine formula

Again from the Cosine Rule, we derive

The Sine Rule for Hyperbolic Triangles
For any h-triangle ABC,
sin(A)/sinh(a) = sin(B)/sinh(b) = sin(C)/sinh(c).

proof of the sine rule

Noting that sin(A) = 1 if A = π/2, we have

The Sine Formula for Hyperbolic Triangles
If the h-triangle ABC has a right angle at A, then
sin(B) = sinh(b)/sinh(a), and
sin(C) = sinh(c)/sinh(a).

The Sine and Cosine Formulae allow us to derive

The Tangent Formula for Hyperbolic Triangles
If the h-triangle ABC has a right angle at A, then
tan(B) = tanh(b)/sinh(c), and
tan(C) = tanh(c)/sinh(b).

proof of the tangent formula

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
For any h-triangle ABC,
sin(B)sin(C)cosh(a) = cos(A) + cos(B) cos(C),
with similar formulae for cosh(b) and cosh(c).

proof of the second cosine rule

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