Your friend isn't stupid, just mathematically transcendent. This right triangle contains the golden ratio (φ ≈ 1.618...) hidden in plain sight. The hypotenuse (3.80422...) divided by the base (2) equals 1.902... which is... completely meaningless. The angle is 58.28°, also irrelevant. And that 1+√5 expression? That's actually the formula for φ, but it equals 3.236... not whatever they're claiming here.
So yes, your friend is absolutely wrong. This is just a random triangle with arbitrary numbers slapped on it by someone who failed both geometry and Greek mythology. The only "golden" thing here is the comedy.