How is the range of angles for which rainbows can be observed calculated?

i. Angle of deviation for the final emergent ray, can be shown to be equal to δ = π + 2i – 4r for primary rainbow and δ = 2π + 2i – 6r for the secondary rainbow.

ii. Using these relations along with Snell’s law, sin i = n sin r, derivatives of angle of deviation (δ) is obtained.

iii. Second derivative of δ comes out to be negative, which shows that it is the minima condition.

iv. Equating first derivative to zero corresponding values of i and r are obtained.

Thus, from the figures shown, the corresponding angles θ_R and θ_V at the horizontal are obtained. These angles give the visible angular position for the rainbow.