Accurate Computation

Question

When looking for $x$-values for which $f(x) =1 - \sin(x)$ cannot be accurately computed, we have to look for values the operands are very close in value; in this case this happens when $x = \frac{\pi}{2}$. Why is it then, when evaluating the function for a larger number such as $10^{27}$ the calculator gives an error? Does this means that we cannot accurately computed for large $x$-values, too?