$\begin{align}
& \left\{ \begin{matrix}
\Phi =AB\cos \omega t\to \cos \omega t=\frac{\Phi }{AB}=\frac{\Phi }{{{\Phi }_{m}}} \\
I={{I}_{m}}\sin \omega t\to \sin \omega t=\frac{I}{{{I}_{m}}} \\
\end{matrix}\xrightarrow{({{\cos }^{2}}\omega t+{{\sin }^{2}}\omega t=1)} \right.(\frac{\Phi }{{{\Phi }_{m}}})+(\frac{I}{{{I}_{m}}})=1 \\
& (\left| \Phi \right|=\frac{1}{2}{{\Phi }_{m}}\leftrightarrow \left| \frac{\Phi }{{{\Phi }_{m}}} \right|=\frac{1}{2})\to {{(\frac{1}{2})}^{2}}+{{(\frac{I}{{{I}_{m}}})}^{2}}=1\to {{(\frac{I}{{{I}_{m}}})}^{2}}=1-\frac{1}{4}=\frac{3}{4}\to \left| \frac{I}{{{I}_{m}}} \right|=\sqrt{\frac{3}{4}}=\frac{\sqrt{3}}{2} \\
\end{align}$