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 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

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=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