Are polygonometric functions related to trigonometric functions for all angles?

[Loren Booda]

Place the center of a regular polygon with number of sides s=3, 4, 5... at an origin. Also, let its vertexes intersect a unit circle, with one vertex at zero radians.

Define functions similar to those trigonometric. The sweeping vector out from the origin now varies in both its arc around from zero radians as well as its corresponding distance to the side of the polygon.

I believe these "polygonometric" functions are continuous. As s-->oo, the polygonometric functions approach the trigonometric. The relation of polygonometric functions with finite s to trigonometric functions may demonstrate great utility.

Has this idea been introduced before? What might be some of the applications for polygonometric functions?

 

[Eynstone]

Such functions would obviously be continuous & approach the trigonometric. However, they can be expressed in terms of trigonometric functions conveniently & I'd have two functions {sin,cos} rather than an infinity of polygonometrics.

 

[Loren Booda]

How would they be expressed in terms of trigonometric functions?

 

[Eynstone]

We can find where a ray making an angle 't' will cut the regular n-gon. The x & y coordinates will be the n-gonometric functions; expressible in terms of sint ,cost and n.

 

[Loren Booda]

We can find where a ray making an angle 't' will cut the regular n-gon. The x & y coordinates will be the n-gonometric functions; expressible in terms of sint ,cost and n.

Isn't that essentially what I've already said? What I'm asking is whether there is an algebraic relationship between the nth polygonometric function and its analogous trigonometric function over all angles.