MORE THAN YOU EVER WANTED TO KNOW ABOUT SWITCH FROG NUMBERS AND SWITCH FROG ANGLES

How does one calculate the frog number using measurements at the track?

Given the angle of divergence at the frog: how does one calculate the frog number?

Given the frog number: how does one calculate the angle of divergence?

Summary

Calculate the frog number (“N”)
Visualize a line of symmetry through the center of the frog with the rails diverging on either side of that line. Pick a point x units down the line from the point of the frog and measure, perpendicular to the center line, left or right to the rail; call that measurement y units. The SPREAD is 2y; and N = x/2y. Those of you who don’t have the time to wade through the following detail sections can skip to the Summary. 
Given the angle of divergence at the frog, calculate the frog number
Call the frog angle “alpha”. The cotangent of alpha/2 is x/y (the adjacent side of the right triangle divided by the opposite side), and x/y = 2N:
cotan (alpha/2) = x/y = 2N
N = cotan (alpha/2) / 2.
Or using tangent = 1/cotangent:
tan (alpha/2) = y/x = 1/cotan (alpha/2)
N = cotan (alpha/2) / 2
N = (1 / tan (alpha/2)) / 2
N = 1/(2 tan (alpha/2))
(In case you don’t have cotangent on your hand calculator, it is the inverse of the tangent–1/tan; tan is opposite over adjacent; cotan is adjacent over opposite. The cotan of alpha is also tan (90alpha). If you have an ancient and honorable analog computer called a “slide rule”, you probably have a cotangent “table” on it.)
Wait! There’s more! Actually, it turns out that the inverse of the sine of alpha
1/sin (alpha)
is very close to the frog number!
N = 1/sin(alpha)

Given the frog number, calculate the angle of divergence
From 1) above, N = x/2y so y/x = 1/2N.
The tangent of alpha/2 is also y/x. So
tan (alpha/2) = y/x = 1/2N,
alpha/2 = arctan (1/2N),
alpha = 2 arctan (1/2N).
As a check, solve 2) above for alpha:
N = 1/(2 tan (alpha/2)),
tan (alpha/2) = 1/2N,
alpha/2 = arctan (1/2N),
alpha = 2 arctan (1/2N).
Alternatively, for a close approximation, solve from 2) above:
N = 1/sin (alpha)
for alpha, getting:
alpha = arcsin (1/N)

Summary
Exact:
N = x/2y
N = 1/(2 tan (alpha/2))
alpha = 2 arctan (1/2N).
Close Enough:
N = 1/sin (alpha)
alpha = arcsin (1/N)