Output of system given input and frequency response
Euler's formula is you friend :) $$e^{j\theta} = \cos(\theta) + j\sin(\theta)$$
After applying it, you will have $$H(e^{j\omega}) = \Re\{H(e^{j\omega})\} + j\Im\{H(e^{j\omega})\}$$ and then it's easy to find magnitude $$|H(e^{j\omega})|=\sqrt{\Re^2\{H(e^{j\omega})\} + \Im^2\{H(e^{j\omega})\}}$$ and phase $$\angle H(e^{j\omega}) = \tan^{-1}\frac{\Im\{H(e^{j\omega})\}}{\Re\{H(e^{j\omega})\}} $$