We're talking about the sine addition formulas. Right now I want to do an example that uses the sine of a sum formula and if you recall the sine of a sum is the sine of alpha plus beta equals and the way to remember the sine of a sum formula is sine cosine cosine sine, sine alpha cosine beta, cosine alpha sine beta and unlike cosine, the cosine formulas, the sine formulas have the same sign, plus or minus, so think 's' for same.
Now this formula, this expression here is exactly in the form sine, cosine sine so I can use the sine of a sum formula in reverse where alpha is 5 pi over 12 and beta is pi over 12. So this expression becomes sine of 5 pi over 12 plus pi over 12 which is the sine of 6 pi over 12 or pi over 2. What's the sine of pi over 2? 1. So this whole expression simplifies to just 1.
I have another example; evaluate the sine of 105 degrees. Now here we want to try to find exact values of possible and it is possible because 105 degrees can be expressed as 60 degrees plus 45 degrees, so sine of 60 degrees and 45 degrees and I can use the sine of a sum. Sine, cosine, cosine, sine and because it's a plus I have a plus in the middle, just fill in the values and you should know these are all special values of sine and cosine.
Sine of 60 degrees root 3 over 2, cosine of 45 root 2 over 2, cosine of 60 is 1/2 and sine of 45 degrees root 2 over 2, so you get root 6 over 4 plus root 2 over 4 which is root 6 plus root 2 all over 4 and that's your answer. Remember sine, cosine, cosine, sine, plus stays plus.