SOLUTION: Use a half-angle formula to find the exact value of tan 3pi/8

Question 515214: Use a half-angle formula to find the exact value of tan 3pi/8
Answer by drcole(72) (Show Source): You can put this solution on YOUR website!
The half angle formula for tangent can be written in two different but equally simple ways:

We'll use the first formula, but you could just as easily use the second and get the same answer. We want to find the tangent of . The angle is one half of the angle , since:

So we'll try applying the half angle formula for tangent using .

Using the unit circle, we can see that and (this makes sense since the angle is in the second quadrant and makes a 45 degree angle with the negative x-axis). So we substitute in:

Simplifying, we get:
(canceling the negatives in the numerator)
(getting rid of the complex fraction by multiplying by the reciprocal of the denominator instead)
(distributing the )
(simplifying to be )
So the tangent of is exactly. This should match what you get on your calculator, and it also makes sense: is in the first quadrant, so its tangent should be positive, and it is larger angle than 45 degrees, so its tangent should be larger than 1.
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