SOLUTION: what is the exact value of tan 3pi/16

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Question 386784: what is the exact value of tan 3pi/16
Answer by robertb(5830) About Me  (Show Source):
You can put this solution on YOUR website!
Find the exact value of tan%28%283pi%29%2F16%29.
Recall the identity tan%282x%29+=+%282tanx%29%2F%281-tan%5E2+%28x%29%29.
Then from this come
(1)-----> tan%28%283pi%29%2F8%29+=+%282tan%28%283pi%29%2F16%29%29%2F%281-tan%5E2%283pi%2F16%29%29, and
(2)-----> tan%28%283pi%29%2F4%29+=+%282tan%28%283pi%29%2F8%29%29%2F%281-tan%5E2%283pi%2F8%29%29. Let alpha+=+tan%28%283pi%29%2F8%29
From (2) we get -1+=+%282alpha%29%2F%281-alpha%5E2%29.
From this we get alpha%5E2+-+2alpha+-+1+=+0. This has solutions
alpha+=+%282+%2B-+sqrt%288%29%29%2F2+=+1+%2B-+sqrt%282%29, using the quadratic formula.
Eliminate 1-sqrt%282%29, because alpha+=+tan%28%283pi%29%2F8%29 >0.
Then substitute this value into (1):
, letting beta+=++tan%28%283pi%29%2F16%29.
Hence the quadratic equation %281%2Bsqrt%282%29%29beta%5E2+%2B+2beta+-+%281%2Bsqrt%282%29%29+=+0.
The solutions to this equation are beta+=+%28-1+%2B-+sqrt%284%2B2sqrt%282%29%29%29%2F%281%2Bsqrt%282%29%29. Again eliminate the negative beta value. Hence beta+=+%28+sqrt%284%2B2sqrt%282%29%29+-+1%29%2F%281%2Bsqrt%282%29%29, the exact value of tan%28%283pi%29%2F16%29.