SOLUTION: How do you find the tan of 165 using double/half angle formulas?

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Question 615987: How do you find the tan of 165 using double/half angle formulas?
Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
If 165 is double or half of a special angle then the answer is yes. With a little effort we should be able to answer this with: Yes. 165 = (1/2)(330) and 330 is a special angle. So:
Let x = 330/ Then (1/2)x = 165
Now we can use
tan%28%281%2F2%29x%29+=+%281-cos%28x%29%29%2Fsin%28x%29
Replacing x with 330 and (1/2)x with 165 we get:
tan%28165%29+=+%281-cos%28330%29%29%2Fsin%28330%29

An angle of 330 degrees terminates in the 4th quadrant and has a reference angle of 30 degrees.
Since sin is negative in the 4th quadrant and since sin(30) = 1/2, sin(330) = -1/2.
Since cos is positive in the 4th quadrant and since cos(30) = sqrt%283%29%2F2, cos(330) = sqrt%283%29%2F2.
Substituting these into the formula we get:
tan%28165%29+=+%281-%28sqrt%283%29%2F2%29%29%2F%28-1%2F2%29
Multiplying the top and bottom of the fraction by 2 will eliminate the "little" fractions:
tan%28165%29+=+%28%281-%28sqrt%283%29%2F2%29%29%2F%28-1%2F2%29%29%282%2F2%29
which simplifies to:
tan%28165%29+=+%282-sqrt%283%29%29%2F-1
tan%28165%29+=+-2%2Bsqrt%283%29
This is the exact value for tan(165).