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 approximation with csc(x) = c/a and cot(x) = b/a. and you have
x ≈ 3a/(b + 2c)
in radians. Multiply by 180°/π to convert to degrees, and note that 540/π ≈ 172.
[1] J. S. Frame. Solving a right triangle without tables. The American Mathematical Monthly, Vol. 50, No. 10 (Dec., 1943), pp. 622-626