How do you tell whether the value of cos [(pi)/4] is positive, negative, zero or undefined?

Out of $6$ trigonometric function $\sin x$ and $\cos x$ are the only two defined for all real values of $x$.
To find if the value is positive, negative or zero you have to know in which quadrant the angle lies. pi/4 in (0;pi/2), so it lies in the first quadrant. For this quadrant all trigonometric functions are positive.