A transparent solid cylindrical rod has a refractive index . . .

Question :  A transparent solid cylindrical rod has a refractive index of 2/√3. It is surrounded by air. A light ray is incident at the midpoint of one end of the rod as shown in the figure. The incident angle θ for which the light ray grazes along the wall of the rod is :

a)sin^-1(1/2)
b) sin^-1(√3/2)
c) sin^-1(2/√3)
d) sin^-1(1/√3)

Doubt by Aishwarya

Solution : 

Refractive index of transparent solid cylindrical rod w.r.t. air (µ₂₁) = 2/√3

So

µ₁₂ =1/µ₂₁ 

= 1/(2/√3) 

= √3/2

µ₁₂ = √3/2

At Normal N1'N2'

µ₂₁=1/sinic
2/√3=1/sinic
sini_c=1/(2/√3)
sini_c=√3/2
sini_c=sin60°
i_c=60°

In Right Triangle 
r₁+90°+i_c=180°(ASP)
r₁+90°+60°=180°
r₁+150°=180°
r₁=180°-150°
r₁=30°

At Normal N1N2
Using Snell's Law

µ₂₁=sini₁/sinr₁
2/√3=sinθ/sin30°
(2/√3)×sin30° = sinθ
(2/√3)×(1/2) =sinθ
(1/√3) = sinθ
sinθ=1/√3
θ = sin^-1(1/√3)

Hence, d) would be the correct option.