Feature of quaternion Cos, Cot etc

Question

I am new to SE Mathematica, and search gives no answer; so here is my problem:

During the testing of my numerical library for quaternions with Mathematica (package Quaternions) I observed the following behavior:

<<Quaternions`
a = Quaternion[1, 2, 3, 4]
(* Quaternion[1, 2, 3, 4] *)

b = N[ArcCos[a]]
(* Quaternion[1.390115947, -0.8918751006, -1.337812651, -1.783750201] *)

Cos[b]
(* Quaternion[1., -2., -3., -4.] *)

i.e. the vector part is negated. I can reproduce the result for ArcCos, but my code gives Cos[b] = [1, 2, 3, 4] (also given by the quaternion postfix calculator). This is not a rounding problem, and not limited to this specific quaternion. The same happened for e.g. Cot[ArcCot[a], where as the following is as I would expect:

b = N[ArcSin[a]]
(* Quaternion[0.1806803802, 0.8918751006, 1.337812651, 1.783750201] *)

Sin[b]
(* Quaternion[1., 2., 3., 4.] *)

My first guess was that it is related to branch cuts, but the analog complex functions give the expected result Cos[ArcCos[1 + 2*I]] = 1 + 2*I.

Question: What is the explanation for the negated vector part for some trigonometric quaternion functions like $\cos$, $\cot$, etc.?

Answer 1

It does seem to be a matter of branch cut choice. If I use a different definition for quaternionic ArcCos[] (e.g. the definition in Morais et al.'s book):

<<Quaternions`

qq = N[Quaternion[1, 2, 3, 4]];

Sign[qq - Re[qq]] ** Log[qq + Exp[Log[qq ** qq - 1]/2]]
   Quaternion[-1.39012, 0.891875, 1.33781, 1.78375]

compared with

ArcCos[qq]
   Quaternion[1.39012, -0.891875, -1.33781, -1.78375]

the former yields qq when Cos[] is applied to it, while the latter has the behavior shown in the OP.

We have here the amusing conclusion that in general $\cos q\ne\cos(-q)$ for quaternionic $q$.