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User:LucasVB/Square sine and cosine functions

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The square sine along with the common sine

The square sine and square cosine functions are akin to their trigonometric counterparts, but instead of defining an unit circle, they define a square of "radius" 1 (that is, side 2). I'm not sure if such functions are already properly defined in the mathematical community, but I never heard of them. I doubt I'm the first to toy with this concept, though.

The square sine ("sinsk") can be written as:

The square cosine ("cosk") is defined as:

The function gives the radius for a n-sided polygon at the angle x. In other words, is the "polar polygon function". N-gon sine/cosine functions are analogous. As n increases, the functions will approach the circular sine and cosines.

Approximations

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An interesting approximation can be done by using iterated trigonometric functions:

Define a function ts such as:

The square sine can then be approximated by:

Which gives a smooth curve that differs no more than 0.1082300356377... from the square sine. I wonder if there's a better approximation...