What is the angle of incidence in air of a light ray whose angle of refraction in glass is half the angle of incidence? show proof and calculations.

θ₁ = cos⁻¹ (n₁ / 2n₂)

Explanation:

For this exercise let's use the law of refraction

       n₁ sint θ₁ = n₂. sin θ₂

Where n₁ and n₂ are the refractive indices for the two media, θ₁ and θ₂, the angles of incidence and refraction

They tell us that the angle of incidence is equal to the angle refracted over 2

    θ₁ = θ₂ / 2

    θ₂ = 2 θ₁

Let's replace

    n₁ sin θ₁ = n₂ sin θ₂

Let's use the trigonometry relationship

    sin 2θ = 2 sinθ cos θ

    n₁ sin θ₁ = n₂ (2 sin θ₁ cos θ₁)

    n₁ = n₂ cos θ₁

    cos θ₁ = n₁ / 2 n₂

    θ₁ = cos⁻¹ (n₁ / 2n₂)

Therefore, the angle of incidence is

   θ₁ = cos⁻¹ (n₁ / 2n₂)

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