Apply Newton's trick of generalizing the Bionomial expansion to real powers:
\sin(rx) :=\sum_{k=0}^\infty {r \choose k} \cos^k x \sin^{r-k} x \sin[(r-k)\frac{\pi}{2}]
Alireza Jamali's Blog
To create tetration, it is better that we work with a commutative exponentiation
x\wedge y:=x^{\ln y}.
With this definition, tetration would be the solution to the functional equation
f(ab)=f(a)\wedge f(b)={f(a)}^{\ln f(b)}
Properties
1. f(1)=e
2.
\ln f(\frac{1}{x})=\frac{1}{\ln f(x)},
so
3. \ln f(-1)=\pm 1