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It is known that in a right triangle with angles 30 and 60 degrees the cathetus at the 60 angle is equal to the 0.5 of hypotenuse. In other words an angle with cosine 0.5 is equal to PI/3. Is there any way to find (using only four rational arithmetic operations) other ratios which are integer multiples to PI, so that if we have a unit radius (hypotenuse) and a cosine which equals to some exact rational number (e.g. 0.00025(0)) then there is a way to know that this angle is some integer multiple to PI?

Albert
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    I've seen several questions/answers pointing to the fact that if $\theta$ is an algebraic multiple of $\pi$ then $\cos \theta$ is an algebraic number. See here for example. – abiessu Feb 23 '14 at 01:46
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    You mean rational multiple of $\pi$. The rational multiples of $\pi$ whose $\cos$ is rational are the integer multiples of $\pi/2$, and the relatives of your $\pi/3$, namely $2n\pi \pm \pi/3$. That's all. – André Nicolas Feb 23 '14 at 02:17
  • @AndréNicolas Perhpas you could give a link to the proof. – Sawarnik Feb 23 '14 at 10:43

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