A monochromatic light wave with wavelength λ _ 1 and frequency ν _...

By using Snell's law, ${n}_{1}sin{\theta}_{1} = {n}_{2}sin{\theta}_{2}$, the refractive index is

$\mu = \dfrac{\sin 45 °}{\sin 30 °} = \dfrac{\dfrac{1}{\sqrt{2}}}{\dfrac{1}{2}} = \sqrt{2}$

Also using the relation $n = \dfrac{c}{v}$ where $n$ is the refractive index, $c$ is the velocity of light in air and $v$ is the velocity in the medium

$⇒ \dfrac{{c}_{\text{air}}}{\text{v}} = \sqrt{2} = \dfrac{{\lambda}_{\text{1}}}{{\lambda}_{2}}$

The velocities are given by

$v = ν{\lambda}_{2}c = ν{\lambda}_{1}$

$⇒ {\lambda}_{1} = \sqrt{2}{\lambda}_{2}$ and ${ν}_{1} = {ν}_{2}$.

The frequency of light does not change when the medium changes.