Angle Sums and Pythagorean Triples

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 formula for sine. If is the smaller acute angle in a 3-4-5 triangle, then

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 Pythagorean triples!

For example, suppose is an acute angle in a right triangle with . Then

By the Pythagorean identity

And so

which of course also follows directly from algebra.

For example, using this process

Originally posted on Mastodon.

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