Signature of $\pi$

We do not know how to compute the signature of $\pi$ within arguments of trigonometrics. $\pi$ plays a very specific role in trigonometric functions. So far all attempts to find $S_n(\pi)$ so that the trigonometric functions compute correctly yield values which make too many mistakes. The reason for the apparent failures to find a signature are illustrated in the following. Since $\sin ( k \pi ) = 0$ for integer k and $\sin( e\pi )$ is reduced to $\sin( S_n(e) S_n(\pi) )$ and all signatures are integers, the signature of sin of any expression times $\pi$ becomes 0. In other words, the condition k is an integer is not unique when we are dealing with signatures, all expressions are mapped to integers. New ideas are needed to resolve this problem.