Suppose you wish to obtain the exact value of sin(165).  We know exact values of the following angles on the unit circle : 0, 30, 45, 60, 90, 120, 135, 150,. 180, 210, 225, 240, 270, 300, 315, 330, 360. 

If we pick two angles from the unit circle that add to 165, we can obtain the exact value for sin(165). The reason is we can substitute those values in for this formula.

sin(a + b) = sin(a)cos(b)+cos(a)sin(b)

Notice that 120 + 45 = 165 and both of those values are on the unit circle

sin(120 + 45)= sin(120)cos(45) + cos(120)sin(45)

                     = (sqrt(3)/2)(sqrt(2)/2) + (-1/2)(sqrt(2)/2)

                     = sqrt(6)/4  - sqrt(2)/4

                    =  (sqrt(6) - sqrt(2))/4