Answer
$$s = \frac{{2\pi }}{3}$$
Work Step by Step
$$\eqalign{
& \left[ {\frac{\pi }{2},\pi } \right];{\text{ tan }}s = - \sqrt 3 \cr
& {\text{Using the tangent definition}} \cr
& {\text{tan }}s = \frac{{\sin s}}{{\cos s}} = - \sqrt 3 \cr
& {\text{From the unit circle we can see that in the interval }}\left[ {\frac{\pi }{2},\pi } \right],{\text{ }} \cr
& {\text{the arc length }}s = \frac{{2\pi }}{3}{\text{ is associate with the point }}\left( {\frac{{\sqrt 3 }}{2}, - \frac{1}{2}} \right). \cr
& {\text{The first coordinate is }} \cr
& {\text{cos }}s = \cos \frac{{2\pi }}{3} = - \frac{1}{2} \cr
& {\text{The second coordinate is }} \cr
& \sin s = \sin \frac{{2\pi }}{3} = \frac{{\sqrt 3 }}{2} \cr
& \cr
& {\text{tan }}s = \frac{{\sin \left( {2\pi /3} \right)}}{{\cos \left( {2\pi /3} \right)}} = \frac{{\frac{{\sqrt 3 }}{2}}}{{ - \frac{1}{2}}} = - \sqrt 3 \cr
& {\text{then }}s = \frac{{2\pi }}{3} \cr} $$
