Question 1032966
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Use the sum or difference identities to find the exact value of the trigonometric function.
tan([23pi]/[12])
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<pre>
{{{tan(23pi/12)}}} = {{{tan(-pi/12)}}} = {{{-tan(pi/12)}}} = -tan(15°) = {{{-sin(15^o)/cos(15^o)}}}.

sin(15°) = {{{sqrt((1-cos(30^o))/2)}}} =   (formula of half argument for sine, see the lesson  <A HREF=https://www.algebra.com/algebra/homework/Trigonometry-basics/Trigonometric-functions-of-half-argument.lesson>Trigonometric functions of half argument</A>  in this site)

= {{{sqrt((1 - sqrt(3)/2)/2)}}} = {{{sqrt((2-sqrt(3))/4)}}} = {{{sqrt(2-sqrt(3))/2}}}.


cos(15°) = {{{sqrt((1+cos(30^o))/2)}}} =   (formula of half argument for sine, see the lesson  <A HREF=https://www.algebra.com/algebra/homework/Trigonometry-basics/Trigonometric-functions-of-half-argument.lesson>Trigonometric functions of half argument</A>  in this site)

= {{{sqrt((1 + sqrt(3)/2)/2)}}} = {{{sqrt((2+sqrt(3))/4)}}} = {{{sqrt(2+sqrt(3))/2}}}.


Therefore,  tan(15°) = {{{sqrt(2-sqrt(3))/sqrt(2+sqrt(3))}}}.

You can rationalize the denominator in the last expression by multiplying the numerator and the denominator by {{{sqrt(2-sqrt(3))}}}. You will get 

tan(15°) = {{{sqrt(2-sqrt(3))/sqrt(2+sqrt(3))}}} = {{{sqrt((2-sqrt(3))^2)/(sqrt(2+sqrt(3))*sqrt(2-sqrt(3)))}}} = {{{(2-sqrt(3))/sqrt(4-sqrt(3)^2)}}} = {{{2-sqrt(3)}}}.

Finally,  {{{tan(23pi/12)}}} = {{{-tan(15^o)}}} = {{{-(2-sqrt(3))}}} = {{{sqrt(3)-2}}}  (negative value).

It is your answer. The problem is solved.


See also the lesson  <A HREF=https://www.algebra.com/algebra/homework/Trigonometry-basics/Trigonometric-functions-of-half-argument-Examples.lesson>Trigonometric functions of half argument - Examples</A>,  Example 2. 
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