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    graphing utility to graph

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    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.)

    Please show detailed working.

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    Solution Preview

    Please refer to the attachment for full solution, with diagrams and equations.

    (A) Deflection angle in radians is given as (D) and is given by (1).

    D = &#1087; - 2&#945; - ...

    Solution Summary

    Law of Refraction, Rainbows, and Raindrops are exemplified.