In a recent post I mentioned in passing that trigonometry can be generalized from functions associated with a circle to functions associated with other curves. This post will go into that a little further.

The equation of the unit circle is

and so in the first quadrant

The length of an arc from (1, 0) to (cos θ, sin θ) is θ. If we write the arc length as an integral we have

and so

is the inverse sine of *x*. Sine is the inverse of the inverse of sine, so we could define the sine function to be the inverse of *F*.

This would be a complicated way to define the sine function, but it suggests ways to create variations on sine: take the length of an arc along a curve other than the circle, and call the inverse of this function a new kind of sine. Or tinker with the integral defining *F*, whether or not the resulting integral corresponds to the length along a familiar curve, and use that to define a generalized sine.

## Example: sin_{p}

We can replace the 2’s in the integral above with *p*‘s, defining *F*_{p} as

and defining sin_{p} to be the inverse of *F*_{p}. When *p* = 2, sin_{p}(*x*) = sin(*x*). This idea goes back to E. Lungberg in 1879.

The function sin_{p} has its applications. For example, just as the sine function is an eigenfunction of the Laplacian, sin_{p} is an eigenfunction of the *p*-Laplacian.

We can extend sin_{p} to be a periodic function with period 4*F*_{p}(1). The constants π_{p} are defined as 2*F*_{p}(1) so that sin_{p} has period π_{p} and π_{2} = π.

## Future posts

I intend to explore several generalizations of sine and cosine. What happens if you replace a circle with an ellipse or a hyperbola? Or a squircle? How do these variations on sine and cosine compare to the originals? Do they satisfy analogous identities? How do they appear in applications? I’d like to address some of these questions in future posts.

It’s probably not going to work out well, but could you use this to create trigonometry for the p-adics?

You can define trig functions over the p-adics from the power series, if the power series converges.