Ask Uncle Colin: Rational Trigonometric Values | Colin

Dear Uncle Colin,

You know how sometimes $\sin(2x)$ is rational and $\sin(5x)$ is rational and $\sin(7x)$ is rational, right? Would that necessarily mean that $\sin(12x)$ is rational?

Asking for a friend.

— Perhaps You THink All Geometry’s On Right Angled Stuff

Hi, PYTHAGORAS, I believe it does! (In fact, I can prove it.)

I’m going to use two identities for it:
$\cos(2A) \equiv 1 – \sin^2(A)$
$2\cos(B)\sin(C) \equiv \sin(B+C) – \sin(B-C)$

… as well as the fact that the rationals are closed under the four basic operations1 : if you add, subtract, multiply or divide two rational…

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