Why is base 60 more precise for trigonometry, can you give an example? (opens in new tab)

With decimal numbers we can write 1/3 as 3/10+3/100+3/1000+..., it is infinite series converging to 1/3. With base 60 we can write it as 20/60, just one term of series is not null. When dealing with infinite series one would need to pick few first terms of series dropping the rest thus losing precision.

With decimal numbers we can write down exact quotient when divide by powers of 2 and 5, because 10=2 * 5. With base twelve we could write exact quotients from division by powers of 2 and 3. With base 60 we could write anything that comes from division by powers of 2, 3 and 5. It can be archieved with base 30=2 * 3 * 5, and I'm not sure that base 60 superior to 30, it do not allow to write more fractional numbers precisely. Maybe it sometimes simplifies arithmetic operations though. I believe, it is easier to divide by 4 in base 60 than in base 30.

Now we have processors that can deal with great amount of terms of series and can keep any reasonable precision we need (in the most cases at least), but back then it was hard work for skilled enough to deal with arithmetic operations.

Counting to twelve is easy with one hand. Counting to sixty is easy with two hands. It is a modern thing to count to ten with two hands.