Question 612149
I assume that the problem is to find an exact value for {{{tan(pi/12)}}}. Please post the entire problem, including the instructions.<br>
When the phrase "exact value" is used in Trig problems, it is code for "put away your calculator and use special angles".<br>
But {{{pi/12}}} is not a special angle!? So how an we do this? The answer is: Find a way to express {{{pi/12}}} in terms of special angles and then use appropriate Trig properties to find the tan of that expression.<br>
The special angles are 0, {{{pi/6}}}, {{{pi/4}}}, {{{pi/3}}},{{{pi/2}}}, and multiples of these. SO we want to find some expression involving these special angles that is equal to {{{pi/12}}}. There are probably other expressions that will work, but one that come to mind is: {{{pi/3-pi/4}}}. This expression, properly subtracted, is equal to {{{pi/12}}}. So we can rewrite
{{{tan(pi/12)}}}
as
{{{tan(pi/3-pi/4)}}}
Now we can use {{{tan(A-B) = (tan(A)-tan(B))/(1 + tan(A)tan(B))}}}, to rewrite this in terms of {{{tan(pi/3)}}} and {{{tan(pi/4)}}}:
{{{(tan(pi/3) - tan(pi/4))/(1+tan(pi/3)*tan(pi/4))}}}<br>
Since {{{tan(pi/3) = sqrt(3)}}} and {{{tan(pi/4) = 1}}}, this expression becomes:
{{{((sqrt(3)) - (1))/(1+(sqrt(3))*(1))}}}
which simplifies to:
{{{(sqrt(3) - 1)/(1+sqrt(3))}}}
This may be an acceptable answer. But it does have a square root in the denominator so you may want to rationalize it:
{{{((sqrt(3) - 1)/(1+sqrt(3)))((1-sqrt(3))/(1-sqrt(3)))}}}
{{{(sqrt(3)-3-1+sqrt(3))/(1^2 - (sqrt(3))^2)}}}
{{{(2sqrt(3) - 4)/(1-3)}}}
{{{(2sqrt(3) - 4)/(-2)}}}
{{{(-2(-sqrt(3) +2))/(-2)}}}
{{{(cross(-2)(-sqrt(3) +2))/(cross(-2))}}}
{{{-sqrt(3) +2}}}
or 
{{{2-sqrt(3)}}}<br>
So the exact value for {{{tan(pi/12)}}}, in simplest form, is {{{2-sqrt(3)}}}