Vrftsjtryk

I've got a linear time-invariant system $$y[n]=frac{8}{9}y[n-1]+x[n]$$
which i transformed into a transfer function $$Y(z)=frac{8}{9}Y(z)*z^{-1}+X(z) =>frac{Y(z)}{X(z)}=frac{1}{1-frac{8}{9}*z^{-1}}=frac{z}{z-frac{8}{9}}$$
The pole is positive, which would make the system stabile. However, i am having some trouble getting the magnitude/phase responses |H(ω)| and θ(ω). How should i proceed with the system to figure these out? I've read about using the substitution for z as$$Y(z)=frac{z}{z-frac{8}{9}}=frac{e^{iw}}{e^{iw}-frac{8}{9}}$$
or even continue to transform the exponentials to their cosine + sin forms, but i am not sure how i should proceed with them. I've thought of something like this:
$$|H(ω)|=frac{z}{z-frac{8}{9}}=frac{e^{iw}}{e^{iw}-frac{8}{9}}=frac{sqrt{cos(ω)^2+sin(ω)^2}}{sqrt{(cos(ω)-frac{8}{9})^2+sin(ω)^2}}=frac{1}{sqrt{(cos(ω)-frac{8}{9})^2+sin(ω)^2}}$$

As an addition, how would the system change if i would to swap x[n] with a sin or a cosine of some value? I assume it would be the same path towards the responses but with different values.