Patrick Honner on Nostr: I've always found it cool that if you double the smaller acute angle in a 3-4-5 ...

I've always found it cool that if you double the smaller acute angle in a 3-4-5 triangle you get the larger acute angle in a 7-24-25 right triangle. You can see this as a consequence of the double angle for sine. If \( \alpha \) is the smaller acute angle in a 3-4-5 triangle, then

\[ \sin (2\alpha) = 2\sin\alpha\cos\alpha=2\frac{3}{5}\frac{4}{5}=\frac{24}{25}\]

In fact, if the sine and cosine of an angle are both rational, then so will be the sine and cosine of twice that angle. This gives a way to turn Pythagorean triples into new Pythaogorean triples!

For example, suppose \( \alpha \) is an acute angle in a right triangle with \(a^2 + b^2 = c^2 \). Then

\[ \sin 2\alpha = \frac{2ab}{c^2} \]
\[ \cos 2 \alpha = \frac{a^2-b^2}{c^2} \]

By the Pythagorean identity

\[ \left(\frac{2ab}{c^2} \right)^2 + \left(\frac{a^2-b^2}{c^2} \right)^2 = 1\]

And so

\[ \left(2ab \right)^2 + \left(a^2-b^2 \right)^2 = \left(c^2\right)^2\]
which of course also follows directly from algebra.

For example, using this process

\[ (3,4,5) \mapsto (7,24,25) \mapsto(336,527,625)\]

#Math #MathEd #Mathematics #NumberTheory #Algebra #Geometry