SOLUTION: Use the angle sum identities to find the exact value of the trigonometric expression tan 345° A. 2 = sqrt3 B. -2 - sqrt3 C. -2 + sqrt3 D. 2 - sqrt3

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Question 627579: Use the angle sum identities to find the exact value of the trigonometric expression tan 345°
A. 2 = sqrt3

B. -2 - sqrt3

C. -2 + sqrt3

D. 2 - sqrt3

Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
When you read or hear "exact value" you should learn that it means "put your calculator down". These problems can and should be solved by hand. Often the solution will involve special angles whose function values should be well known.

We cannot find tan(345) by hand directly since the reference angle for 345 is 15 and 15 is not a special angle. But by using a sum, difference, double or half angle formula we can find tan(345) if it is
  • a sum of special angles
  • a difference of special angles
  • twice a special angles
  • half a special angles
With a little effort you should be able to find a combination that works. There are numerous possibilities:
345 = 210 + 135 (for which we would use tan%28A%2BB%29=%28tan%28A%29%2Btan%28B%29%29%2F%281-tan%28A%29tan%28B%29%29)
345 = 390 - 45 (for which we would use tan%28A-B%29=%28tan%28A%29-tan%28B%29%29%2F%281%2Btan%28A%29tan%28B%29%29)
345 = (1/2)690 (for which we would use
%281-cos%28690%29%29%2Fsin%28690%29
Since cos%28690%29+=+sqrt%283%29%2F2 and sin%28690%29+=+-1%2F2:
%281-sqrt%283%29%2F2%29%2F%28-1%2F2%29
Eliminating the fractions within a fraction:
%28%281-sqrt%283%29%2F2%29%2F%28-1%2F2%29%29%2A%282%2F2%29
%282-sqrt%283%29%29%2F%28-1%29
which simplifies to:
-2+%2B+sqrt%283%29

You're welcome to try one of the other combinations of special angles that are equal to 345 and see if you get this answer.