Find the domain of $f(x)=\log_{10}(1-\log_7(x^2-5x+13))+\arccos\left(\frac{3}{2+\sin\frac{9\pi x}{2}}\right)$ Find the domain of definition of $$f(x)=\log_{10}(1-\log_7(x^2-5x+13))+\arccos\left(\frac{3}{2+\sin\frac{9\pi x}{2}}\right)$$

I found the domain of $\log_{10}(1-\log_7(x^2-5x+13))$ to be $x\in(2,3)$ and the domain of $\arccos\left(\frac{3}{2+\sin\frac{9\pi x}{2}}\right)$ to be $\sin\frac{9\pi x}{2}=1$ i.e. $x=1,5,9$ etc but the answer given is $\frac{21}{9},\frac{25}{9}$.I dont know where i am wrong.

Since $\sin(\frac{9\pi x}{2})=1$,
$$\frac{9\pi x}{2}\in(\frac{\pi}{2}, \frac{5\pi}{2}...\frac{21\pi}{2}...) \quad or \quad x\in(\frac{1}{9}, \frac{5}{9},..., \frac{21}{9}, \frac{25}{9}...)$$
We have seen that the domain of the first part of equation should be in $(2,3)$, So the only feasible solution is $\frac{21}{9}$ and $\frac{25}{9}$.
