(Too long for a comment.)

It's a bit older than your reference, but so-called "hypergoniometric functions" have been considered by Erik Lundberg in 1879. This article is a more recent discussion. Shelupsky and Burgoyne discuss similar generalizations. All ultimately consider this as the problem of inverting an appropriate generalization of the integral representations of arcsine and arccosine.

The $n=3$ case has been considered separately by A.C. Dixon; I had talked a bit about Dixon elliptic functions in this math.SE answer.

(Too long for a comment.)

It's a bit older than your reference, but so-called "hypergoniometric functions" have been considered by Erik Lundberg in 1879. This article is a more recent discussion. Shelupsky and Burgoyne discuss similar generalizations. All ultimately consider this as the problem of inverting an appropriate generalization of the integral representations of arcsine and arccosine.

The $n=3$ case has been considered separately by A.C. Dixon; I had talked a bit about Dixon elliptic functions in this math.SE answer.