 Solution

In the $e^{i\omega t}$ phasor convention, the total series RL circuit impedance is $R+i\omega L$. The resistor voltage follows the current, so its transfer function relative to the applied voltage isAt the stated condition $R=\omega L$,Thus the resistor-voltage amplitude is $\boxed{V_{R,0}=V_{\mathrm{in},0}/\sqrt{2}}$ and it lags by $45^{\circ }$. For a cosine input it is $V_R(t)=V_{\mathrm{in},0}\cos (\omega t-\pi /4)/\sqrt{2}$.