Well, we have the following circuit:
If the input voltage is given by the following time depended function:
$$\text{V}_\text{in}\left(t\right)=\hat{\text{v}}\cos\left(\omega t+\varphi\right)\tag1$$
If we write that voltage in a complex way, we get:
$$\underline{\text{V}}_{\space\text{in}}=\hat{\text{v}}\exp\left(\varphi\text{j}\right)\tag2$$
Where $\text{j}^2=-1$.
The input impedance is given by:
$$\underline{\text{Z}}_{\space\text{in}}=\text{R}+\text{j}\omega\text{L}\tag3$$
So, the complex input current is given by:
$$\underline{\text{I}}_{\space\text{in}}=\frac{\underline{\text{V}}_{\space\text{in}}}{\underline{\text{Z}}_{\space\text{in}}}=\frac{\hat{\text{v}}\exp\left(\varphi\text{j}\right)}{\text{R}+\text{j}\omega\text{L}}\tag4$$
So, the time dependent input current is given by:
$$\text{I}_\text{in}\left(t\right)=\left|\underline{\text{I}}_{\space\text{in}}\right|\cos\left(\omega t+\arg\left(\underline{\text{I}}_{\space\text{in}}\right)\right)\tag5$$
Where:
$$\left|\underline{\text{I}}_{\space\text{in}}\right|=\frac{\hat{\text{v}}}{\sqrt{\text{R}^2+\left(\omega\text{L}\right)^2}}\tag6$$
$$\arg\left(\underline{\text{I}}_{\space\text{in}}\right)=\varphi-\arctan\left(\frac{\omega\text{L}}{\text{R}}\right)\tag7$$
