SOLUTION: Evaluate: sin 2pi cos pi/3 - cos 2pi sin pi/3 P.S. the answer is -square root of 3/2 but I don't know the right solution

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Question 677589: Evaluate: sin 2pi cos pi/3 - cos 2pi sin pi/3
P.S. the answer is -square root of 3/2 but I don't know the right solution
Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
sin%282pi%29cos%28pi%2F3%29+-+cos%282pi%29sin%28pi%2F3%29
One way to do this takes advantage of the fact that all the angles in the expression are special angles. You should know (or learn) that:
sin%282pi%29+=+0
cos%28pi%2F3%29+=+1%2F2
cos%282pi%29+=+1
sin%28pi%2F3%29+=+sqrt%283%29%2F2
Substituting these values into your expression:
%280%29%2A%281%2F2%29+-+%281%29%28sqrt%283%29%2F2%29
which simplifies to:
-sqrt%283%29%2F2

Another way to do this uses the fact that the expression matches the pattern of the right side of:
sin%28A-B%29+=+sin%28A%29cos%28B%29-cos%28A%29sin%28B%29
with the "A" being 2pi and the "B" being pi%2F3. Using the formula we can rewrite your expression as sin(A-B):
sin%282pi-pi%2F3%29
We should know that the angle 2pi-pi%2F3 is an angle that terminates in the 4th quadrant and that its reference angle is pi%2F3. The sin of the reference angle is sqrt%283%29%2F2 and since sin is negative in the 4th quadrant:
sin%282pi-pi%2F3%29+=+-sqrt%283%29%2F2