I did the calculation manually, the answer is exactly 10 divided by 14. Basically you just take ( 1 * 3 + 2 * 2 + 3 * 1 ) / (3 * 3 + 2 * 2 + 1 * 1), which is 10 / 14. Computing cosine similarities is really simple as I said.

Edit: The funny thing is that his answer is actually a bit too low since he truncates instead of rounds at the end. 10 / 14 just continues as 142857 and repeats after that. So if the library is smaller than his then the library is wrong and not him. Anyway, blindly trusting the results of a library is just dumb, they make mistakes so often you are right and they are wrong.