User:LucasVB/Square sine and cosine functions

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:

${\mbox{sinsq}}(x)={\mbox{PP}}_{4}(x)\sin(x)$

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

${\mbox{cossq}}(x)={\mbox{PP}}_{4}(x)\cos(x)$

The function ${\mbox{PP}}_{n}(x)=\sec({\tfrac {2}{n}}\sin ^{-1}(\sin({\tfrac {n}{2}}x)))$ gives the radius for a n-sided polygon at the angle x. In other words, ${\mbox{PP}}_{n}(x)$ 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

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

Define a function ts such as:

${\mbox{ts}}_{0}(x)=x$

${\mbox{ts}}_{1}(x)=\tan(\sin(x))$

${\mbox{ts}}_{n}(x)=\tan(\sin({\mbox{ts}}_{n-1}(x)))$

The square sine can then be approximated by:

${\mbox{sinsq}}(x)\approx {\frac {2}{\pi }}{\mbox{ts}}_{7}(x)$

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