The refractive index of a medium is inversely proportional to the speed of light in that medium, given by n=vc.
For light travelling from medium A to medium B, the critical angle θc is defined when light travels from a denser to a rarer medium.
Given speeds are vA=2.4×108 m/s and vB=2.7×108 m/s.
The refractive indices are nA=vAc and nB=vBc.
The critical angle θc is given by sinθc=nAnB=vBvA.
Substituting the values: sinθc=2.7×1082.4×108=2724=98.
To find the value in terms of tan−1, we use the trigonometric identity for a right-angled triangle where perpendicular p=8 and hypotenuse h=9.
The base b=h2−p2=92−82=81−64=17.
Thus, tanθc=bp=178.
Therefore, θc=tan−1(178).