document.write( "Question 612149: tan pie/12
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Algebra.Com's Answer #385298 by jsmallt9(3758) $\"\"$ $\"About$

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I assume that the problem is to find an exact value for $\"tan%28pi%2F12%29\"$ . Please post the entire problem, including the instructions.

\n" ); document.write( "When the phrase \"exact value\" is used in Trig problems, it is code for \"put away your calculator and use special angles\".

\n" ); document.write( "But $\"pi%2F12\"$ is not a special angle!? So how an we do this? The answer is: Find a way to express $\"pi%2F12\"$ in terms of special angles and then use appropriate Trig properties to find the tan of that expression.

\n" ); document.write( "The special angles are 0, $\"pi%2F6\"$ , $\"pi%2F4\"$ , $\"pi%2F3\"$ , $\"pi%2F2\"$ , and multiples of these. SO we want to find some expression involving these special angles that is equal to $\"pi%2F12\"$ . There are probably other expressions that will work, but one that come to mind is: $\"pi%2F3-pi%2F4\"$ . This expression, properly subtracted, is equal to $\"pi%2F12\"$ . So we can rewrite
\n" ); document.write( " $\"tan%28pi%2F12%29\"$
\n" ); document.write( "as
\n" ); document.write( " $\"tan%28pi%2F3-pi%2F4%29\"$
\n" ); document.write( "Now we can use $\"tan%28A-B%29+=+%28tan%28A%29-tan%28B%29%29%2F%281+%2B+tan%28A%29tan%28B%29%29\"$ , to rewrite this in terms of $\"tan%28pi%2F3%29\"$ and $\"tan%28pi%2F4%29\"$ :
\n" ); document.write( " $\"%28tan%28pi%2F3%29+-+tan%28pi%2F4%29%29%2F%281%2Btan%28pi%2F3%29%2Atan%28pi%2F4%29%29\"$

\n" ); document.write( "Since $\"tan%28pi%2F3%29+=+sqrt%283%29\"$ and $\"tan%28pi%2F4%29+=+1\"$ , this expression becomes:
\n" ); document.write( " $\"%28%28sqrt%283%29%29+-+%281%29%29%2F%281%2B%28sqrt%283%29%29%2A%281%29%29\"$
\n" ); document.write( "which simplifies to:
\n" ); document.write( " $\"%28sqrt%283%29+-+1%29%2F%281%2Bsqrt%283%29%29\"$
\n" ); document.write( "This may be an acceptable answer. But it does have a square root in the denominator so you may want to rationalize it:
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\n" ); document.write( " $\"%28sqrt%283%29-3-1%2Bsqrt%283%29%29%2F%281%5E2+-+%28sqrt%283%29%29%5E2%29\"$
\n" ); document.write( " $\"%282sqrt%283%29+-+4%29%2F%281-3%29\"$
\n" ); document.write( " $\"%282sqrt%283%29+-+4%29%2F%28-2%29\"$
\n" ); document.write( " $\"%28-2%28-sqrt%283%29+%2B2%29%29%2F%28-2%29\"$
\n" ); document.write( " $\"%28cross%28-2%29%28-sqrt%283%29+%2B2%29%29%2F%28cross%28-2%29%29\"$
\n" ); document.write( " $\"-sqrt%283%29+%2B2\"$
\n" ); document.write( "or
\n" ); document.write( " $\"2-sqrt%283%29\"$

\n" ); document.write( "So the exact value for $\"tan%28pi%2F12%29\"$ , in simplest form, is $\"2-sqrt%283%29\"$ \n" ); document.write( "

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