1. The Calculus of Rainbows
Ariella, Sebby, Erando, Isabella, Romain

2. 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.

3. D(α)= (α-β) + (π-2β) + (α-β) = π + 2α-4β

4. 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

5. 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.

6. Calculations In the big triangle, AZC In the small triangle, ABC
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

7. 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

8. 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

9. 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)/ Sin(α)/1.3318=sin(β) Sin-1 (sinα)/ = β D(α)=π+2α-4β 180+2α-4(Sin-1 (sinα)/1.3318) Find the rainbow angle by using the calculator: =42.3  this proves that the rainbow angle is 42.3 for the color red

10. Part 2 continued Sin(α)=k(sinβ) Sin(α) =1.3435sin(β) 
Sin(α)/1.3435=sin(β) Sin-1 (sinα)/ = β
D(α)=π+2α-4β 180+2α-4(Sin-1 (sinα)/1.3435)
Find the rainbow angle by using the calculator:
=40.6  this proves that the rainbow angle is for the color violet