How do I find the exact value without any decimals?
11.25 degrees = pi/16
$\displaystyle \cos 2\theta = 2 \cos^2 \theta - 1 \Rightarrow \cos \theta = \sqrt{\frac{1 + \cos 2\theta}{2}}$ (in the domain that we're interested in)
Then just apply this half-angle formula a couple times to get your answer.