As incident vector \(A\) makes \(i\) angle with normal \(z-\)axis refracted vector \(R\) makes \(r\) angle with normal \(z\) - axis with help of direction cosine

\(i=\cos ^{-1}\left(\frac{A_{z}}{A}\right)=\cos ^{-1}\left(\frac{5}{\sqrt{(4 \sqrt{3})^{2}+(3 \sqrt{3})^{2}+5^{2}}}\right)\)

\(=\cos ^{-1}\left(\frac{5}{10}\right) \Rightarrow i=60^{\circ}\)

\(\sqrt{2} \sin 60=\sqrt{3} \times \sin r\)

\(r=45^{\circ}\)

Difference between \(i \) & \(r =60-45=15\)