Hyperbolic Trigonometry

Trigonometry is the study of the relationships among sides and angles of a triangle. In Euclidean geometry we use similar triangles to define the trigonometric functions--but the theory of similar triangles in not valid in hyperbolic geometry. Thus, we have defined them in terms of their power series expansion for any real number. (See Chapter 7)

Define two hyperbolic trigonometric functions   

Since converges for all , the power series expansions of the hyperbolic trigonometric functions are

and converge for all real x. In fact, using complex analysis and letting , we can easily see that

 
Figure 21.1: Graphs of hyperbolic functions

We get to choose our unit of measurement in -- not a particular model now. We would like to choose a unit so that k=1. It can be shown that this is analogous to the choice of a unit of angle measurement so that a right angle has radian measure  . By measuring in radians the theorem on area becomes 

and the Fundamental Formula  becomes

A straightforward calculation using double angle formulas for the circular functions gives the following formulas:

For example, to derive the first equation:

This function provides a connection between the hyperbolic and circular functions.

Given , let , , and . First, we will derive some formulæ of hyperbolic geometry from the Poincaré model. Choose our circle to have radius 1. Let O denote the center of and let . Let be the Poincaré distance from O to B and let t=d(O,B) be the Euclidean distance from O to B. From Lemma 7.4 we have

We then get that

where F is the isomorphism of the Poincaré model onto the Klein model described in the previous chapter.

Theorem 20.1:   Given any right triangle in the hyperbolic plane and k=1 with the right angle (having measure , then
   

Before we prove these equations, compare them with the formulæ for a right triangle in Euclidean geometry.

1. Equation 21.6 is the hyperbolic analogue of the Pythagorean theorem. 
   
   Thus, if is sufficiently small so that higher powers of a, b, and c are negligible, then
   
   
2. Equation 21.6 says that for sufficiently small and . How close are these approximations? Consider right triangles with fixed and let c approach 0. Since a<c, .
   
   where . Thus,
   
   Thus,
   
   
3. Equation 21.6 and the second equality in Equation 21.6 have no euclidean parallels for there the angles do not determine the lengths of the sides.

Proof: We may assume that A=O is the center of . Let B'=F(B) and C'=F(C). Then

From our above observation we have

The best way to see this is to think of t as t=t+0i, so thatt sits on the x-axis and is sent to a point on the x-axis. The point to which it is sent must be at a distance F(t) from the origin.

It is then easy to see that

Now, let . By our previous results on the inversion of circles, B'' is the inverse of B in . Thus,

Likewise, .

Let be the center of . Let be the foot of on . so that is the midpoint of BB''. Thus, if is tangent to at B, . Since they are opposite angles, . Thus, . Then

Since is an arbitrary acute angle in a right triangle we can relabel to ge that

To prove 21.6

Then,



follows from what we have done above.

Theorem 20.2:   For any triangle in the hyperbolic plane, with k=1

Let us take a look at a specific example. Consider equilateral triangles. In Euclidean geometry all are similar, since they all must have angles measuring . If this were true in hyperbolic geometry, they would have to be congruent by AAA. What then are the angles in an equilateral triangle of differing sides? Look at Table 21.1 and see if you can tell what is happening.

 

Sides Radians Degrees

10 0.0135 0.77

5 0.1633 9.35

2.5 0.5359 30.71

1.5 0.7930 45.43

1 0.9188 52.64

.5 1.0122 57.99

.1 1.0458 59.92

Other interesting examples would be how the angles in a right isosceles triangle varies with the sides and what triangle is analogous to the standard 30-60-90 triangle in Euclidean geometry.

 
• Circumference and Area of a Circle