I don't understand the "rational" arguments for trig functions For example look at $\cos(x)$. I imagine that $x$ can sometimes be rational (or integer) or irrational, and sometimes $\cos(x)$ can be rational (or integer), or irrational. 
Under what circumstances is $\cos(x)$ rational (or integer), or irrational, and under which combinations of rational/integer/irrational inputs $x$ can these arise? 
In other words what are the relationships?
 A: if as usual we use radian measure, the only rational value of $\cos x$ at rational $x$ is at $x=0$. Ivan Niven has given a quite accessible proof. 
Integer values are much more obvious. The only integer $x$ for which $\cos x$ is an integer is $x=0$.  
As to for which irrational $x$ we have $\cos x$ rational, probably not much can be said in general. For example, the good old $3$-$4$-$5$ triangle gives us a couple of rational cosines, but there is no nice independent description of the corresponding angles. 
One can describe completely the $x$ of shape $x=r\pi$, where $r$ is  rational,  such that $\cos x$ is rational. Equivalently, we have complete information about the cosine of angles that are rational when measured in degrees. 
It turns out that one can prove that the only rational multiples of $\pi$ whose cosine is rational are the ones we can easily think of: the integer multiples of $\frac{\pi}{2}$, and the $x$ of shape $\pm\frac{\pi}{3}+2n\pi$ and $\pm \frac{2\pi}{3}+2n\pi$. 
Remark: The results mentioned above can be found in Niven's Rational and Irrational Numbers. If you do not have ready access to the book, googling rational values of trigonometric functions niven gives a fair number of hits. 
