Simple approximation for solving a right triangle

Suppose you have a right triangle with sides a, b, and c, where a is the shortest side and c is the hypotenuse. Then the following approximation from [1] for the angle A opposite side a seems too simple and too accurate to be true. In degrees, A ≈ a 172° / (b + 2c). The approximation above only involves simple arithmetic. No trig functions. Not even a square root. It could be carried out with pencil and paper or even mentally. And yet it is surprisingly accurate. If we use the 3, 4, 5 triangle as an example, the exact value of the smallest angle is A = arctan(3/4) × 180°/π ≈ 36.8699° and the approximate value is A ≈ 3 × 172° / (4 + 2×5) = 258°/7 ≈ 36.8571°, a difference of 0.0128°. When the angle is more acute the approximation is even better. Derivation Where does this magical approximation come from? It boils down to the series 2 csc(x) + cot(x) = 3/x + x³/60 + O(x4) where x is in radians. When x is small, x³/60 is extremely small and so we have 2 csc(x) + cot(x) ≈ 3/x. Apply this…