Five angles whose sum is exactly ninety degrees?

To answer this question I needed some nuclear weaponry, sorry!

Let θ₁, θ₂, θ₃, θ₄, θ₅ are the 5 angles listed in order

(i.e. θ₁ = arcsin(12/37), etc.). All of them are acute angles in the following Pythagorean triangles:

#1: (12, 35, 37); #2: (57, 176, 185); #3: (3232, 9945, 10457);

#4: (7735, 23808, 25033); #5: (385468067, 1250801244, 1308850405)

Closer look at the hypotenuses (denominators of your fractions) shows that 37 in #1 is prime and 185 = 5*37 in #2; 10457 in #3 and 25033 in #4 are prime, but 1308850405 = 5 * 10457 * 25033. That was the reason for me to try to add separately θ₁ + θ₂ and θ₃ + θ₄ + θ₅ using the relationships

arcsin(x) + arcsin(y) = arcsin(x√(1 - y²) + y√(1 - x²))

arcsin(x) + arcsin(y) + arcsin(z) = arcsin(x√(1 - y²)√(1 - z²) +

+ y√(1 - z²)√(1 - x²) + z√(1 - x²)√(1 - y²) - xyz)

(valid because θ₁ + θ₂ < π/2, θ₃ + θ₄ + θ₅ < π/2)

http://en.wikipedia.org/wiki/List_of_trigonometric...

So θ₁ + θ₂ = arcsin((12*176 + 57*35)/(37*185)) = arcsin(3/5);

much more efforts finally produced the following:

θ₃ + θ₄ + θ₅ = arcsin((3232 * 23808 * 1250801244 + 7735 * 9945 * 1250801244 + 385468067 * 9945 * 23808 - 3232 * 7735 * 385468067) / (5 * 10457² * 25033²)) = ... = arcsin(4/5)!

Then (θ₁ + θ₂) + (θ₃ + θ₄ + θ₅) = arcsin(3/5) + arcsin(4/5) = π/2 exactly, as the famous Egyptian Triangle (3, 4, 5) shows. So finally we are done.

How on Earth and where do You dig such terrible relationships?

P.S. You are mostly welcome, Rita! Yes, Pythagorean Triangles are a real Universe, but keeping abreast of mathematical software and updating personal knowledge becomes too much for an old man like me. Long Live the WWW!