Answer
$$s = \frac{\pi }{4}$$
Work Step by Step
$$\eqalign{
& \left[ {0,\frac{\pi }{2}} \right];{\text{ cos }}s = \frac{{\sqrt 2 }}{2} \cr
& {\text{From the unit circle we can see that in the interval }}\left[ {0,\frac{\pi }{2}} \right],{\text{ }} \cr
& {\text{the arc length }}s = \frac{\pi }{4}{\text{ is associate with the point }}\left( {\frac{{\sqrt 2 }}{2},\frac{{\sqrt 2 }}{2}} \right). \cr
& {\text{The first coordinate is }} \cr
& {\text{cos }}s = \cos \frac{\pi }{4} = \frac{{\sqrt 2 }}{2} \cr
& {\text{then}}\,\,s = \frac{\pi }{4} \cr} $$
