Greek letters alpha and beta are denoted by a and b respectively in the description below.

Rainbows are formed when light strikes raindrops and is reflected and refracted as shown in the figure in the attached document. The figure shows a cross section of a spherical raindrop. The Law of Refraction states that (sin a) / (sin b) = k where k = 1.33 (for water).

The angle of deflection is given by D = pi + 2a - 4b.

A) Use a graphing utility to graph

D = pi + 2a - 4 sin-inverse(1/k sin a), 0 <= a <= pi/2

B) Prove that the minimum angle of deflection occurs when:

cos a = square root of [(k^2 - 1) / 3]

C) For water, what is the minimum angle of deflection, Dmin ? (The angle (pi - Dmin) is called the rainbow angle.) What value of a produces this minimum angle? (A ray of sunlight that strikes a raindrop at this angle, a, is called a rainbow ray.)

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