### The Calculus of Rainbows

```The Calculus of Rainbows
Ariella, Sebby, Erando, Isabella, Romain
Introduction
Rainbows are created when raindrops scatter sunlight.
We used the ideas of Descartes and Newton to
explain the shape, location, and colors of rainbows.
D(α)= (α-β) + (π-2β) + (α-β) = π + 2α-4β
Question
The diagram in this problem represents the angles
formed by a ray of sunlight entering a raindrop
reflecting and refracting back to the observer.
The equation below represents the desired angle of
deviation after the proper amount of clockwise
rotations has occurred. The goal in the problem is to
prove that these equations are equal.
D(α)= (α-β) + (π-2β) + (α-β) = π + 2α-4β
How We Got There

*Lets call the angle next to β , x.

* Lets call the angle supplementary to
D(α), z.

* α=β+x x=α-β
*Draw a line connecting point A to point
C. The angles formed opposite of C and
A respectively will each be called y.
Calculations
 In the big triangle, AZC

2y+2β+2x+z=180°

X=α-β

2y+2β+2(α-β)+z=180°
 In the small triangle, ABC

2y + 4β= 180°2y=π-4β y=(π-4β)/2

Plug in… 2(π-4β)/2 + 2(α-β) + 2β+z=π
Calculations Return!
Get Rid of Z:

D(α)+z=180

D(α)=180-z

180=2((π-4β)/2)+2(α-β)+2β

D(α)=2((π-4β)/2)+2(α-β)+2β

D(α)=π-4β+2β+2(α-β)

D(α)=π-4β+2α180-4β+2α

(α-β+(π-2β)+(α-β)=π+2α-4β
Let’s Graph!
 D(α)=180+2α-4(sin-1(.75 sinα)(3/4)
Sin-1(.75sinα)=β
 To prove min = 138° (y) when α= 59.4° (x)3
Part 2
Finding the rainbow angle for red and violet using Snell’s law
K=index of refraction
Sin(α)=k(sinβ)
Sin(α) =1.3318sin(β) 
Sin(α)/1.3318=1.3318sin(β)/1.3318
Sin(α)/1.3318=sin(β) Sin-1 (sinα)/1.3318 = β
D(α)=π+2α-4β 180+2α-4(Sin-1 (sinα)/1.3318)
Find the rainbow angle by using the calculator:
180-137.74=42.3  this proves that the rainbow angle is 42.3 for the color red
Part 2 continued
Sin(α)=k(sinβ)
Sin(α) =1.3435sin(β) 
Sin(α)/1.3435=1.3318sin(β)/1.3435
Sin(α)/1.3435=sin(β) Sin-1 (sinα)/1.3435 = β
D(α)=π+2α-4β 180+2α-4(Sin-1 (sinα)/1.3435)
Find the rainbow angle by using the calculator:
180-139.35=40.6  this proves that the rainbow angle is 40.6
for the color violet
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