Answer
$$ - \sqrt 3 $$
Work Step by Step
$$\eqalign{
& \tan \left( { - \frac{{7\pi }}{3}} \right) \cr
& {\text{Use tan}}\left( { - \theta } \right) = - \tan \theta \cr
& = - \tan \left( {\frac{{7\pi }}{3}} \right) \cr
& {\text{Use the identity }}\tan \theta = \frac{{\sin \theta }}{{\cos \theta }} \cr
& - \tan \left( {\frac{{7\pi }}{3}} \right) = \frac{{ - \sin \left( {7\pi /3} \right)}}{{\cos \left( {7\pi /3} \right)}} \cr
& {\text{Write }}\frac{{7\pi }}{3}{\text{ as }}2\pi + \frac{\pi }{3} \cr
& - \tan \left( {\frac{{7\pi }}{3}} \right) = \frac{{ - \sin \left( {2\pi + \pi /3} \right)}}{{\cos \left( {2\pi + \pi /3} \right)}} \cr
& {\text{Then}} \cr
& - \tan \left( {\frac{{7\pi }}{3}} \right) = \frac{{ - \sin \left( {\pi /3} \right)}}{{\cos \left( {\pi /3} \right)}} \cr
& \cr
& {\text{From the unit circle we known that }} \cr
& \sin \frac{\pi }{3} = \frac{{\sqrt 3 }}{2}{\text{ and cos}}\frac{\pi }{3} = \frac{1}{2} \cr
& - \tan \left( {\frac{{7\pi }}{3}} \right) = \frac{{ - \left( {\sqrt 3 /2} \right)}}{{1/2}} \cr
& - \tan \left( {\frac{{7\pi }}{3}} \right) = - \sqrt 3 \cr} $$
