# Polygonometric functions

• Loren Booda
In summary, the conversation discussed the concept of "polygonometric" functions, which are similar to trigonometric functions but vary in both their arc and distance from the origin. It was mentioned that as the number of sides of the polygon approaches infinity, the polygonometric functions approach the trigonometric functions. The potential applications of these functions were also briefly mentioned, as well as the possibility of expressing them in terms of trigonometric functions. The question of whether there is an algebraic relationship between polygonometric and trigonometric functions was also raised.

#### 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?

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.

How would they be expressed in terms of trigonometric functions?

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.

Eynstone said:
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.