You have that the derivatives around $x=0$ of $y=\sin x$ are
$\{ y^{(n)}(0)\}=\{0,1,0,-1 ,\dots \}$
Since you have a multiplicative factor of $\dfrac{\pi }{4}$ this changes to
$\{ y^{(n)}(0)\}=\left\{0,\dfrac{\pi }{4},0,-\dfrac{\pi^3 }{4^3} ,\dots \right\}$
As a consequence you have that the coefficients are
$c_n=\frac{(-1)^n}{(2n+1)!} \left(\frac{\pi}{4}\right)^{2n+1}$
You can readily check that
$\lim \frac{c_{n+1}}{c_{n}}=0$
from which the radius is $\infty$, i.e., the whole extended real line.
As a general result, the convergence radius of the series for
$y=\sin(ax+b)$
is the whole real line.