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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](http://books.google.com/books?id=ov-IlIEo47cC&pg=PA17&source=gbs_toc_r&cad=4#v=onepage&q&f=false).2012-02-20

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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](http://someclassicalmaths.wordpress.com/2009/07/17/nivens-proof-that-the-trigonometric-and-inverse-trigonometric-functions-are-irrational-for-rational-non-zero-arguments/) that presents the proof.2012-02-20