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# Strange Proofs- I

Prove that $$\cos1^{\circ}$$ is irrational.

Is there any proof for it?

Note by Swapnil Das
2 years, 1 month ago

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Here is an elementary solution using strong induction and contradiction.

Assume that $$\cos 1^\circ$$ is rational. Then $$\cos2^\circ=2\cos^2 2^\circ-1$$ is rational. Note that $$\cos(n+1)^\circ+\cos(n-1)^\circ=2\cos 1^\circ\cos n^\circ$$.

Hence by strong induction $$\cos n^\circ$$ is rational for all integers $$n\ge 1$$. But this is clearly a contradiction as $$\cos 30^\circ$$ is trivially irrational. Thus we conclude that $$\cos 1^\circ$$ is irrational.

- 2 years ago

Hint: Use Chebyshev Polynomials.

Staff - 2 years, 1 month ago