<P> The rainbow will occur where the angle φ is maximum with respect to the angle β . Therefore, from calculus, we can set dφ / dβ = 0, and solve for β, which yields </P> <Dl> <Dd> β max = cos − 1 ⁡ (2 − 1 + n 2 3 n) ≈ 40.2 ∘ (\ displaystyle \ beta _ (\ text (max)) = \ cos ^ (- 1) \ left ((\ frac (2 (\ sqrt (- 1 + n ^ (2)))) ((\ sqrt (3)) n)) \ right) \ approx 40.2 ^ (\ circ)). </Dd> </Dl> <Dd> β max = cos − 1 ⁡ (2 − 1 + n 2 3 n) ≈ 40.2 ∘ (\ displaystyle \ beta _ (\ text (max)) = \ cos ^ (- 1) \ left ((\ frac (2 (\ sqrt (- 1 + n ^ (2)))) ((\ sqrt (3)) n)) \ right) \ approx 40.2 ^ (\ circ)). </Dd> <P> Substituting back into the earlier equation for φ yields 2φ ≈ 42 ° as the radius angle of the rainbow . </P>

What is the end of the rainbow called