This is the fourth post in a series on generalizations of sine and cosine.
The first post looked at defining sine as the inverse of the inverse sine. The reason for this unusual approach is that the inverse sine is given in terms of an arc length and an integral. We can generalize sine by generalizing this arc length and/or generalizing the integral.
The first post mentioned that you could generalize the inverse sine by replacing “2” with “p” in an integral. Specifically, the function
is the inverse sine when p = 2 and in general is the inverse of the function sinp. Unfortunately, there two different ways to define sinp. We next present a generalization that includes both definitions as special cases.
Edmunds, Gurka, and Lang  define the function
and define sinp,q to be its inverse.
The definition of sinp at the top of the post corresponds to sinp,q with p = q in the definition of Edmunds et al.
The other definition, and the one we’ll use for the rest of the post, corresponds to sinr,s where s = p and r = (p-1)/p.
This second definition sinp has a geometric interpretation analogous to that in the previous post for hyperbolic functions . That is, we start at (1, 0) and move clockwise along the p-norm circle until we sweep out an area of α/2. When we have swept out that much area, we are at the point (cosp α, sinp α).
When p = 4, the p-norm circle is also known as a “squircle,” and the p-norm sine and cosine analogs are sometimes placed under the heading “squigonometry.”
Previous posts in the series
 David E. Edmunds, Petr Gurka, Jan Lang. Properties of generalized trigonometric functions. Journal of Approximation Theory 164 (2012) 47–56.
 Chebolu et al. Trigonometric functions in the p-norm https://arxiv.org/abs/2109.14036