how can you take tan inverse of 5 when it is bigger than 1?

You cannot take sin inverse 5 because that means 

$$sin\theta=\frac{5}{1}=\frac{opposite}{hypotenuse}$$

but the hypotenuse has to be the longest side so this doesn't make sense!

Hence,

    $$\\So\;\; sin^{-1}x \;\;\mbox{Only exists for }-1\le x\le1\\
\mbox{This is also true for } cos^{-1}x\\\\$$

But

$$tan\theta = \frac{opp}{adj}$$

Either one of these can be the biggest one so of course you can take  $$tan^{-1}5$$   That is perfectly fine.