You don't! There are ways to find those "manually" but they very complicated, tedious, and subject to errror. Even in the years "BC" (before calculators) we did not do those (except possibly square roots), we looked values up in tables. I recommend you talk to your teacher, or whoever is in charge of the exam, about this.

You don't! Not by hand. There are a few "special cases" you could be expected to know, such as or but it would not be reasonable to expect a person to do that by hand in any restricted time. I recommend, again, that you talk with whomever is in charge about exactly what you will be expected to do on this exam. Perhaps they are planning to give you the values you need. Often, part of the test will be "without calculators" and part "with calculators"..So. If I am left with this:

Cos-1(-4/(sqrt(11)sqrt(14))) how can I solve this?

That's because there is NO "relatively simple explanation"!(I understand that this is not a place to provide a complete explanation so a hint to what I can google for or a website link will be enough to get me started)

Any help is appreciated.

I tried to google a little for myself but I did not find a relatively simple explanation)