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1/p is not the cosine of a rational degree marking

Source: 2015 SDMO High School Problem 3

August 24, 2016
trigonometry

Problem Statement

Let pp be an odd prime. Show that 1π⋅cos⁡−1(1p)\frac{1}{\pi}\cdot\cos^{-1}\left(\frac{1}{p}\right) is irrational. (Note: cos⁡−1(x)\cos^{-1}\left(x\right) is defined to be the unique yy with 0≤y≤π0\leq y\leq\pi such that cos⁡(y)=x\cos\left(y\right)=x.)
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