The refractive index of the material of a prism is `sqrt(2)` and the angle of minimum deviation is `30^(@)` . Calculate the value of the refracting angle of the prism.

Explanation

`"Here" " " mu = sqrt(2) , delta_(m) = 30^(@)`
Let the refracting angle of the prism be A
`"We know", mu = sin""(A + delta_(m))/(sin""(A)/(2)) or, sqrt(2) = sin""(A+ 30^(@))/(sin""(A)/(2))`
`or, " " sqrt(2)* sin""(A)/(2) = sin(15^(@) + (A)/(2))`
`= sin15^(@) * cos""(A)/(2)+ cos15^(@) * sin""(A)/(2)`
`or, " " (sqrt(2) - cos15^(@))sin""(A)/(2) = sin15^(@) * cos""(A)/(2)`
`or, " " (sin""(A)/(2))/(sin""(A)/(2))= (sin15^(@))/(sqrt(2) - cos15^(@))`
`"Now", " " sin15^(@) = sin(45^(@) - 30^(@)) = (1)/(sqrt(2) )* (sqrt(3))/(2) - (1)/(sqrt(2))* (1)/(2) = (sqrt(3) - 1)/(2sqrt(2))`
`"Similarly", cos15^(@) = (sqrt(3) + 1)/(2 sqrt2)`
`therefore " " tan""(A)/(2) = ((sqrt(3) - 1)/(2 sqrt(2)))/(sqrt(2) - (sqrt(3) +1)/(2 sqrt(2))) = (sqrt(3) - 1 )/(3 - sqrt(3)) = (1)/(sqrt(3)) = tan30^(@)`
`therefore " " (A)/(2) = 30^(@) or, A = 60^(@)`