No, you are definitely NOT allowed to use calculators. This is the reason many consider the QR section to be so hard. However, most of the questions can be estimated closely without a calculator.
And to answer that specific question...cos(x)=7/4, I don't think it is solvable. adj/hyp=cos(x) and the adjacent cannot be larger than the hypotenuse in a right triangle. Try it in a calculator and you'll see it's not possible.
Note: you do need to know basic trig stuff, like when cos(x)=1, sin(x)=0, ETC...