Answer
$$s = \frac{{7\pi }}{6}$$
Work Step by Step
$$\eqalign{
& \left[ {\pi ,\frac{{3\pi }}{2}} \right];{\text{ }}\sec s = - \frac{{2\sqrt 3 }}{3} \cr
& {\text{Using the secant definition}} \cr
& {\text{sec }}s = \frac{1}{{\cos s}} = - \frac{{2\sqrt 3 }}{3} \cr
& \cos s = - \frac{{\sqrt 3 }}{2} \cr
& {\text{From the unit circle we can see that in the interval }}\left[ {\pi ,\frac{{3\pi }}{2}} \right],{\text{ }} \cr
& {\text{the arc length }}s = \frac{{7\pi }}{6}{\text{ is associate with the point }}\left( { - \frac{{\sqrt 3 }}{2}, - \frac{1}{2}} \right). \cr
& {\text{The first coordinate is }} \cr
& {\text{cos }}s = \cos \frac{{7\pi }}{6} = - \frac{{\sqrt 3 }}{2} \cr
& {\text{sec }}s = \frac{1}{{\cos \left( {7\pi /6} \right)}} = - \frac{{2\sqrt 3 }}{3} \cr
& {\text{then }}s = \frac{{7\pi }}{6} \cr} $$
