document.write( "Question 386784: what is the exact value of tan 3pi/16 \n" ); document.write( "
Algebra.Com's Answer #275428 by robertb(5830)\"\" \"About 
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Find the exact value of \"tan%28%283pi%29%2F16%29\".
\n" ); document.write( "Recall the identity \"tan%282x%29+=+%282tanx%29%2F%281-tan%5E2+%28x%29%29\".
\n" ); document.write( "Then from this come
\n" ); document.write( "(1)-----> \"tan%28%283pi%29%2F8%29+=+%282tan%28%283pi%29%2F16%29%29%2F%281-tan%5E2%283pi%2F16%29%29\", and
\n" ); document.write( "(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\"
\n" ); document.write( "From (2) we get \"-1+=+%282alpha%29%2F%281-alpha%5E2%29\".
\n" ); document.write( "From this we get \"alpha%5E2+-+2alpha+-+1+=+0\". This has solutions
\n" ); document.write( "\"alpha+=+%282+%2B-+sqrt%288%29%29%2F2+=+1+%2B-+sqrt%282%29\", using the quadratic formula.
\n" ); document.write( "Eliminate \"1-sqrt%282%29\", because \"alpha+=+tan%28%283pi%29%2F8%29\" >0.\r
\n" ); document.write( "\n" ); document.write( "Then substitute this value into (1):
\n" ); document.write( ", letting \"beta+=++tan%28%283pi%29%2F16%29\".
\n" ); document.write( "Hence the quadratic equation \"%281%2Bsqrt%282%29%29beta%5E2+%2B+2beta+-+%281%2Bsqrt%282%29%29+=+0\".
\n" ); document.write( "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\".
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