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3. (10 points) (The angle of the rainbow) The figure on the right shows René Descartes' sketch of how a rainbow is formed (Discours de la méthode, 1637). a light ray from the Sun (AB) enters a raindrop at B and undergoes two refractions (at B and D) and a total internal reflection (at C). The purpose of this problem is to show that the angle DEM at which an observer sees the rainbow is approximately 42°. Assume the raindrop is spherical and show that the angle $\alpha$ between the lines AB and DE, which is equal to the angle DEM, is given by $\alpha(\theta) = 4 \sin^{-1}(\sin(\theta)/n) - 2\theta$, where $\theta$ is the angle of incidence and n = 1.33 is the index of refraction of the raindrop. More outgoing rays concentrate near the maximum of the function $\alpha(\theta)$. This maximum angle $\alpha_{max}$ is the angle at which the rainbow appears.

          3. (10 points) (The angle of the rainbow) The figure on the right shows René Descartes' sketch of how a rainbow is formed (Discours de la méthode, 1637). a light ray from the Sun (AB) enters a raindrop at B and undergoes two refractions (at B and D) and a total internal reflection (at C). The purpose of this problem is to show that the angle DEM at which an observer sees the rainbow is approximately 42°. Assume the raindrop is spherical and show that the angle $\alpha$ between the lines AB and DE, which is equal to the angle DEM, is given by $\alpha(\theta) = 4 \sin^{-1}(\sin(\theta)/n) - 2\theta$, where $\theta$ is the angle of incidence and n = 1.33 is the index of refraction of the raindrop. More outgoing rays concentrate near the maximum of the function $\alpha(\theta)$. This maximum angle $\alpha_{max}$ is the angle at which the rainbow appears.

$3. (10 points) (The angle of the rainbow) The figure on the right shows René Descartes' sketch of how a rainbow is formed (Discours de la méthode, 1637). a light ray from the Sun (AB) enters a raindrop at B and undergoes two refractions (at B and D) and a total internal reflection (at C). The purpose of this problem is to show that the angle DEM at which an observer sees the rainbow is approximately 42°. Assume the raindrop is spherical and show that the angle α between the lines AB and DE, which is equal to the angle DEM, is given by α(θ) = 4 sin^-1(sin(θ)/n) - 2θ, where θ is the angle of incidence and n = 1.33 is the index of refraction of the raindrop. More outgoing rays concentrate near the maximum of the function α(θ). This maximum angle αmax is the angle at which the rainbow appears.$

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