I work in a walk-in tutor lab at my university, and one of the other tutors showed me this a couple of years back and changed my life.
First, you make a table that looks like this.
Then, go ahead and add in some nice denominators of 2 in every entry in the table.
Then give yourself some nice square roots on the numerators.
Alright – now we’re going to fill it in. The only value you have to remember is that sin(0)=0. So we put 0 in the numerator for 0 in the sine column. Then we count up as we move down.
Then we do the opposite in the cosine column.
Then we simplify!
And voila – a beautiful unit circle table.