The mathematics of Rainbows (continued)
 
to write the final formula
G = 4 * sin-1( H / n) - 2 * sin-1( H )
If you plot this for water (n = 1.333) you get the graph below.
Graph of G(H)
Now if you know a little calculus you can take the derivative of G with respect to H and find where it has an extrema by setting the derivative to zero and solving for H. If you know calculus you can do this, if you don't, I can't explain it here but the final result is that the angle G is a maximum for
H = ( (4 - n2) / 3 )1/2

And therefore, G has a maximum value given by
G = 4 * sin-1( ( (4 - n2) / 3 )1/2 / n) - 2 * sin-1( ( (4 - n2) / 3 )1/2 )

Using n for water of 1.333 you get
H = 0.8608
G = 42.0 degrees

You can also see that from the graph above. Now you see why it is that if you stand with your back to the sun so you are facing the same direction as the incoming sunlight, you need to look at an angle of 42.0 degrees to see the rainbow. Also note that this maxima in G(H) means that all of the light that enters at near this height comes out at essentially the same angle and so you get more light at this angle than any other.
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