Cheers, I am given the input signal of $x(t) = A \cos(2πft + \frac{\pi}{4})$ which is linear and time invariant, and I am asked to find the output, if I know that, $f = 500hz$ and $|H(f)| = 1, \angle H(f) = \frac{\pi}{4}$
What I know, is that when given an exponential input, I can compute the output as $x(t) = A e^{j\omega_0t}$, then $y(t) = H(\omega_0)x(t)$. $H$ is a function that has an imaginary part, so we can say that: $H(f) = |H(f)|e^{j \arg H(f)}$.Thus I would say that the output is: $1 \times A \cos(2πft + \frac{\pi}{4}) \times e^{j\frac{π}{4}}$
My professor however states that the answer is: $1 \times A \cos(2πft + \frac{\pi}{4} + \frac{\pi}{4}) $, but I can't understand how he simply puts the $\frac{\pi}{4}$ inside the output of cosine. I tried to work it out using the euler's formula, but that doesn't seem to do the trick. Can anyone explain the correct answer? Thanks =)