Trigonometric Functions of Regular Polygons

A circle can be considered as a regular polygon with infinite number of sides. Trigonometric functions can be defined as the lengths of various line segments from a unit circle. In this applet we visualize the trigonometric functions for regular polygons. We see that for low values of n the number of sides in a regular polygon these trigonometric functions deviate from their form for the circle. But, as the number of sides increases they tend towards the circular form. This is true for all the trigonometric functions. This provides yet another way of understanding the circle as a regular polygon with infinite number of sides. The slider n controls the number of sides of the regular polygon. The check boxes can turn a given function ON or OFF. The trigonometric functions for a circle are shown in the background for comparison. To run the animation click on the small triangle at the bottom left of the page.
  1. For low values of n there are certain kinks in the functions? Is there any relation in the number of kinks and n?
  2. What is the threshold value of n above which the regular polygon functions are not distinguishable from circular functions?