Generalized trigonometry

Posted on by John

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

x^2 + y^2 = 1

and so in the first quadrant

y = \sqrt{1 - x^2}

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

\int_0^{\sin \theta} (1 -t^2)^{-1/2} \,dt = \theta

and so

F(x) = \int_0^x (1 - t^2)^{-1/2} \,dt

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: sinp

We can replace the 2’s in the integral above with p‘s, defining Fp as

F_p(x) = \int_0^x (1 - |t|^p)^{-1/p} \,dt

and defining sinp to be the inverse of Fp. When p = 2, sinp(x) = sin(x). This idea goes back to E. Lungberg in 1879.

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

We can extend sinp to be a periodic function with period 4Fp(1). The constants πp are defined as 2Fp(1) so that sinp has period πp and π2 = 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.

2 thoughts on “Generalized trigonometry”

  1. Greg Baker

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

  2. John

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

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