I have a transfer function $$H(z)=\frac{1+1.2z^{-1}+0.8z^{z^-2}}{1-0.9z^{-1}}$$ from which I'm supposed to sketch the magnitude and phase response. I know that you can transform $z=e^{j\omega}$ to get the frequency response, but I don't really understand what it means. The magnitude response is then $$\left|H(e^{j\omega})\right|=\left|\frac{1+1.2e^{-j\omega}+0.8e^{-j2\omega}}{1-0.9e^{-j\omega}}\right|,$$ but how do I calculate the values from here?
For example, if we set $\omega=0$, then the result is $30$. But if we set $\omega=\pi/4$, then the result is $$\left|\frac{1+1.2e^{-j\pi/4}+0.8e^{-j\pi/2}}{1-0.9e^{-j\pi/4}}\right|.$$ How does one solve this?
I also know that the formula for phase response is $$\arg(H(e^{j\omega}))=\frac{\Im(H(e^{j\omega}))}{\Re(H(e^{j\omega}))}$$ but I'm having a hard time to understand how to use it and what phase response actually is. Can you give me an example of how to calculate the phase response of a certain angle?