show that nsini =constant
where n is refractive index and i is angle of incidence
What do you mean "show"?
That is Snell's law
n1 sin theta 1 = n2 sin theta 2
Do you want to prove it by using speed of the wave in the two media or something?
If so, look at the diagram of the two parallel rays here:
https://math.stackexchange.com/questions/153775/simple-proof-for-snells-law-of-refraction
https://math.stackexchange.com/a/2496002
To prove that nsini = constant, where n is the refractive index and i is the angle of incidence, we can use Snell's law and the definition of the refractive index.
Snell's law states that the ratio of the sines of the angles of incidence and refraction is equal to the inverse ratio of the refractive indices of the two media:
n₁sini₁ = n₂sinr₂
where n₁ and n₂ are the refractive indices of the first and second media, respectively, i₁ is the angle of incidence, and r₂ is the angle of refraction.
Now, assume that the angle of incidence and the refractive indices are constant. Let's say that n₁ = n and i₁ = i.
Using Snell's law, we can simplify the equation:
n₁sini₁ = nsinr₂
Since we assume that n₁ = n and i₁ = i, we have:
nsini = nsinr₂
But according to the law of reflection, the angle of incidence is equal to the angle of reflection (i = r). Therefore, we can rewrite the equation as:
nsini = nsini
Thus, nsini is a constant, as the equation demonstrates the equality of the initial angle of incidence and the refractive index.