David Shelupsky [1] suggested a generalization of sine and cosine based on solutions to the system of differential equations

with initial conditions α_{s}(0) = 0 and β_{s}(0) = 1.

If *s* = 2, then α(*t*) = sin(*t*) and β(*t*) = cos(*t*). The differential equations above reduce to the familiar fact that the derivative of sine is cosine, and the derivative of cosine is negative sine.

For larger even values of *s*, the functions α_{s} and β_{s} look like sine and cosine respectively, though flatter at their maxima and minima. Numerical experiments suggest that the solutions are periodic and the period increases with *s*. [2]

Here’s a plot for *s* = 4.

The first zero of α(*t*) is at 3.7066, greater than π. In the plot *t* ranges from 0 to 4π, but the second period isn’t finished.

If we look at the phase plot, i.e (α(*t*), β(*t*)), we get a shape that I’ve blogged about before: a squircle!

This is because, as Shelupsky proved,

## Odd order

The comments above mostly concern the case of even *s*. When *s* is odd, functions α_{s} and β_{s} don’t seem much like sine or cosine. Here are plots for *s* = 3

and *s* = 5.

## Other generalizations of sine and cosine

[1] David Shelupsky. A Generalization of the Trigonometric Functions. The American Mathematical Monthly, Dec. 1959, pp. 879-884

[2] After doing my numerical experiments I looked back more carefully at [1] and saw that the author proves that the solutions for even values of *s* are periodic, and that the periods increase with *s*, converging to 4 as *s* goes to infinity.

Do the curves for odd s look like chunky log and exp? (I wouldn’t be shocked.)

Thanks for the post! To get a sine/cosine-like solution for any real s >= 1, change the DEs to da/dt = sign(b) abs(b)^(s-1), db/dt = -sign(a) abs(a)^(s-1).

Also, I think the period goes from 4 to 8 as s goes from 1 to infinity (tested numerically). And the phase plot morphs from a diamond to a square.