TITLE: Why $\arccos(\frac{1}{3})$ is an irrational number?
QUESTION [11 upvotes]: 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.

REPLY [15 votes]: 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.