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I have just started taking signal & systems lessons and here is my question:

If we say that x0(t) = A.cos(w0t). (a cosine signal with zero phase shift, w0 radian frequency) And if we would like to time-shift this signal by for example t1.

The resulting phase shift should be found by -w0*t1 = phi. (phi denoting phase shift).

By this logic, if I wanted to find the phase-shift for this signal: x(t) = sin(pi/4*t-pi/4)

What would the result be? Can I apply the first formula for phase for already phase shifted signals? Or am I missing something very big here? :)

Thank you very much for any kind of help.

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Indeed, you can apply the same technique:

As you said, given a sine signal with time shift $t_1$, which is $x(t)=\sin(w(t-t_1))$, this becomes $x(t)=\sin(w t - wt_1)=\sin(wt+\phi)$ and $\phi=-wt_1$.

Now, if you encounter a signal $x(t)=\sin(\frac{\pi}{4} t - \frac{\pi}{4})$ you identify $w=\frac{\pi}{4}$ and $\phi=-\frac{\pi}{4}$. What remains is to solve $\phi=-w t_1$ for $t_1$ and you have it.

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