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I was reading the following question. It is a very nice question with a very nice answer! I would like to know why $\arccos(\frac{1}{3})$ is an irrational number.

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See here. – David Mitra Feb 20 '12 at 14:19
up vote 12 down vote accepted

This is a consequence of a result of Ivan Niven: The trigonometric functions are irrational at non-zero rational values of the arguments.

Particularly, your question is addressed by the corollary to the above (Corollary 2.8 in the link): any non-zero value of an inverse trigonometric function is irrational for a rational value of the argument.

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See also this blog post that presents the proof. – joriki Feb 20 '12 at 14:28

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