\(\sin(rx)\) for real r

$$\sin(nx)=\sum_{k=0}^n {n \choose k} \cos^k x \sin^{n-k} x \sin[(n-k)\frac{\pi}{2}]$$

Apply Newton's trick of generalizing the Bionomial expansion to real powers:

$$\sin(rx) :=\sum_{k=0}^\infty {r \choose k} \cos^k x \sin^{r-k} x \sin[(r-k)\frac{\pi}{2}]$$