(Solved): Express \( f \) in terms of unit step functions. \[ \begin{array}{l} f(t)=\sin (t-\pi) \mathscr{U} ...

Express \( f \) in terms of unit step functions. \[ \begin{array}{l} f(t)=\sin (t-\pi) \mathscr{U}(\pi-3 \pi) \\ f(t)=\sin (t) \mathscr{U}(t-\pi)-\sin (t-3 \pi) \\ f(t)=\sin (t+\pi) \mathscr{U}(t-\pi)-\sin (t+3 \pi) \mathscr{U}(t-3 \pi) \\ f(t)=\sin (t)-\sin (t) \mathscr{U}(t-\pi)+\sin (t) \mathscr{U}(t-3 \pi) \\ f(t)=-\sin (t-\pi) \mathscr{U}(t-\pi)+\sin (t-3 \pi) \mathscr{U}(t-3 \pi) \end{array} \] Find \( \mathscr{L}\{f(t)\} \) and \( \mathscr{L}\left\{e^{t} f(t)\right\} \). (Enter your answer in terms of \( s \). ) \( \mathscr{L}\{f(t)\}= \) \[ \mathscr{L}\left\{e^{t} f(t)\right\}= \]