12.9: Hyperbolic trigonometry
( \newcommand{\kernel}{\mathrm{null}\,}\)
In this section we give formulas for h-distance using hyperbolic functions. One of these formulas will be used in the proof of the hyperbolic Pythagorean theorem (Theorem 13.6.1).
Recall that cosh, sinh, and tanh denote hyperbolic cosine, hyperbolic sine, and hyperbolic tangent; that is, the functions defined by
coshx:=ex+e−x2, sinhx:=ex−e−x2,
tanhx:=sinhxcoshx.
These hyperbolic functions are analogous to sine and cosine and tangent.
Prove the following identities:
cosh′x=sinhx; sinh′x=coshx; (coshx)2−(sinhx)2=1.
The identities
cosh(2⋅x)=(coshx)2+(sinhx)2 and sinh(2⋅x)=2⋅sinhx⋅coshx
hold for any real value x.
- Proof
-
(sinhx)2+(coshx)2=(ex−e−x2)2+(ex+e−x2)2==e2⋅x+e−2⋅x2==cosh(2⋅x);
2⋅sinhx⋅coshx=2⋅(ex−e−x2)⋅(ex+e−x2)=e2⋅x−e−2⋅x2=cosh(2⋅x).
Let P and Q be two h-poins distinct from the center of absolute. Denote by P′ and Q′ the inverses of P and Q in the absolute.
(a) cosh[12⋅PQh]=√PQ′⋅P′QPP′⋅QQ′;
(b) sinh[12⋅PQh]=√PQ⋅P′Q′PP′⋅QQ′;
(c) tanh[12⋅PQh]=√PQ⋅P′Q′PQ′⋅P′Q;
(d) coshPQh=PQ⋅P′Q′+PQ′⋅P′QPP′⋅QQ′.
- Hint
-
By Corollary 10.6.1 and Theorem 10.2.1, the right hand sides in the identities survive under an inversion in a circle perpendicular to the absolute.
As usual we assume that the absolute is a unit circle. Suppose that O denotes the h-midpoint of [PQ]h. By the main observation (Theorem 12.3.1) we can assume that O is the center of the absolute. In this case O is also the Euclidean midpoint of [PQ]. (Instead, we may move Q to the center of absolute. In this case the derivations are simpler. But since Q′Q=Q′P=QP=∞, one has to justify that ∞∞=1 every time.)
Set a=OP=OQ; in this case we have
PQ=2⋅a,P′Q′=2⋅1a, PP′=QQ′=1a−a,PQ′=QP′=1a+a.
and
PQh=ln(1+a)2(1−a)2=2⋅ln1+a1−a.
Therefore
cosh[12⋅PQh]=12⋅(1+a1−a+1−a1+a)==1+a21−a2; √PQ′⋅P′QPP′⋅QQ′=1a+a1a−a==1+a21−a2.
Hence the part (a) follows. Similarly,
sinh[12⋅PQh]=12⋅(1+a1−a−1−a1+a)==2⋅a1−a2; √PQ⋅P′Q′PP′⋅QQ′=21a−a==2⋅a1−a2.
Hence the part (b) follows.
The parts (c) and (d) follow from (a), (b), the definition of hyperbolic tangent, and the double-argument identity for hyperbolic cosine, see Theorem 12.9.1.
