sqrt 3 times sin30 - cos30 = sqrt 3 over 2 - sqrt 3 over 2 = 0
but when i press this in my calculator, it appears -5 x 10^(-12). why is that?
Your calculator uses an algorithm to determine what the value of the trig functions are. It doesn't know that , it knows that or so. So sometimes instead of giving the correct answer it will give a close approximation.
The only time you would ever notice this is when you calculate something close to 0: then scientific notation takes over and you can see some really small answers.
Trig functions aren't the only things that can cause this. Square roots, logarithms, etc. Anything that the calculator needs to approximate.