further trigonometry for hyperbolic geometry

For any real θ, the values of sin(θ), cos(θ) and tan(θ) are related.
For any real x, the values of sinh(x), cosh(x) and tanh(x) are related.
If a hyperbolic triangle has angles α,β,γ, and sides a,b,c, then the various rules give us some
relations between the trigonometric functions of the angles and the hyperbolic lengths of the sides.
We can also derive formulae involving the sizes of other angles and lengths aσociated with such
a triangle. The identities allow us to recast these formula in many ways. In the literature, a formula
may appear in quite different looking forms.

Here, we introduce some important triangle invariants and show how these are related, both to one
another, and to quantities such as the inradius, circumradius, and exradii. We do not claim to give a
comprehensive list, or to produce the "best" formulae in any sense.

the function Δ
For complex a,b,c, we define
Δ2 = Δ2 (a,b,c)= 1 + 2 cosh(a)cosh(b)cosh(c) - cosh2(a) - cosh2(b) - cosh2(c), and
s = s(a,b,c) = ½(a+b+c).
(Δ1) Δ2 = 4sinh(s)sinh(s-a)sinh(s-b)sinh(s-c),
(Δ1') Δ2 = sinh2(a)sinh2(b)-(cosh(a)cosh(b)-cosh(c))2, etc.
the function Φ
For complex α,β,γ
Φ2 = Φ(α,β,γ) = -1 + 2cos(α)cos(β)cos(γ)+cos2(α)+cos2(β)+cos2(γ), and
σ = σ(α,β,γ) = ½(α+β+γ).

(Φ1) Φ2 = 4cos(σ)cos(σ-α)cos(σ-β)cos(σ-γ),
(Φ1') Φ2 = sin2(α)sin2(β)-(cos(α)cos(β)+cos(γ))2, etc.

These two functions are closely related.

some Δ, Φ relations
(ΔΦ) If sin(α)sin(β)sin(γ) ≠ 0, let cosh(a) = (cos(β)cos(γ)+cos(α))/sin(α)sin(β), etc
Then (sin(α)sin(β)sin(γ))2Δ2= Φ4.
(ΦΔ) If sinh(a)sinh(b)sinh(c) ≠ 0, let cos(α) = (cosh(b)cosh(c)-cosh(a))/sinh(a)sinh(b), etc
Then (sinh(a)sinh(b)sinh(c))2Φ2= Δ4.

These are all formal consequences of the identities cos2(θ)+sin2(θ) = 1, 2cos(θ)cos(φ) = cos(θ+φ)+cos(θ-φ),
2sin(θ)sin(φ) = cos(θ-φ)-cos(θ+φ) and their hyperbolic analogues.

further hyperbolic triangle formulae
Consider the triangle ABC with a = d(B,C), b = d(C,A), c = d(A,B), α = <CAB, β = <ABC, γ = <BCA,
(Δ2) Δ2 > 0, so we may choose Δ > 0,
(Φ2) Φ2 > 0, so we may choose Φ > 0,
(R1) Δ = Φ2/sin(α)sin(β)sin(γ),
(R2) Φ = Δ2/sinh(a)sinh(b)sinh(c),
(R3) ΔΦ = sin(α)sin(β)sin(γ)sinh(a)sinh(b)sinh(c),
(R4) Δ3 = sin(α)sin(β)sin(γ)(sinh(a)sinh(b)sinh(c))2,
(R5) Φ3 = (sin(α)sin(β)sin(γ))2sinh(a)sinh(b)sinh(c).
(Δ3) Δ = sinh(a)sinh(b)sin(γ) = sinh(b)sinh(c)sin(α) = sinh(c)sinh(a)sin(β).
(Φ3) Φ = sin(α)sin(β)sinh(c) = sin(β)sin(γ)sinh(a) = sin(γ)sin(α)sinh(b).


The reader may well have observed that the formulae occur in pairs, except for (R3).
Indeed, some of our proofs made use of the fact the algebra for (Φn) is almost the
same as that for (Δn). Only the combination of the sum and difference of cosines or
hyperbolic cosines leads to the different look of (Δ1) and (Φ1).

On a purely algebraic level, we can relate the pairs by the following substitutions

  • replace cos(α) by -cosh(a) and cosh(a) by -cos(α), etc,
  • replace sin(α) by -i.sinh(a) and, sinh(x) by i.sin(α), etc,
  • replace Φ by -i.Δ, and Δ by i.Φ.
Note that this is to be applied only to formulae which involve functions of a single
length or angle. Terms involving sums can be dealt with by using the appropriate
identities. For example :
cosh(a+b) = cosh(a)cosh(b)+sinh(a)sinh(b), so is replaced by
(-cos(α))(-cos(β))+(i.sin(α))(i.sin(β)) = cos(α+β).
The reader may care to check that sinh(a+b) is replaced by -i.sin(α+β), and hence that
cosh(a+b+c) is replaced by -cos(α+β+γ).
We can also include functions of half a length or angle. For example
2cosh2(½a) = cosh(a)+1, so is replaced by -cos(α)+1 = 2sin2(½α).
The reader may care to check that 2sinh2(½a) is replaced by -2cos2(½α).
Care is needed in extracting the appropriate root, but since lengths are positive,
and ½α is acute, we can deduce the correct sign.

This is more than a coincidence. If we apply the process to the First Cosine Rule,
we get the Second. If we apply it to the Sine Rule, we get an equivalent form.
It follows that, if we apply the process to a result obtained by using these rules,
then we get another valid result. This shows that the pairs above are logically
equivalent, and (R3) is unaltered.

We shall refer to this process as duality. Thus, (R1) and (R2) are dual theorems,
and (R3) is self-dual. If we recall that hyperbolic geometry has a projective model
based on a projective conic C, then we can apply projective duality with respect to C
to get a map relating points and lines. The basic property of a pair of points is distance,
the basic property of a pair of lines is angle, so we should expect these to be related.
Of course, this is not a proof that we get the above realisation of the duality.

Our first application of duality is to prove a result about the semi-perimeter of a
hyperbolic triangle.

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