Amplitude and Phase angle

Knowing that $A\cos(\omega t - \phi) = A[\cos(\omega t)\cos(\phi) + \sin(\omega t)\sin(\phi)]$ , we obtain:

$A\cos(\phi) =4$ ;

$A\sin(\phi) =1$

and $\frac{4}{\cos(\phi)} = A = \frac{1}{\sin(\phi)} \Rightarrow \tan{\phi} = \frac{1}{4} \Rightarrow \phi = \tan^{-1}(\frac{1}{4}).$ Solving for $A$ yields:

$A = \frac{1}{\sin(\phi)} = \csc(\phi) = \csc(\tan^{-1}(1/4)) = \csc(\csc^{-1}(\sqrt{17})) = \sqrt{17}.$

Finally, we compute the product $A\phi = \sqrt{17} \cdot \tan^{-1}(1/4) = \sqrt{17} \cdot \frac{1}{\sqrt{17}} = \boxed{1}.$