Answer
$x\approx ^+_- .59705$
Work Step by Step
Find the values of c that are guaranteed by the mean value theorem
$\int ^{\frac{\pi}{3}}_{-\frac{\pi}{3}} cosxdx= cosx(\frac{\pi}{3} + \frac{\pi}{3})$, Integrate
$[sinx] ^{\frac{\pi}{3}}_{-\frac{\pi}{3}} = \frac{2\pi}{3}cosx$
$(\frac{\sqrt 3}{2}+\frac{\sqrt 3}{2}) = \frac{2\pi}{3}cosx$
$\frac{3\sqrt 3}{2\pi}= cosx$
$x=cos^{-1}\frac{3\sqrt 3}{2\pi}$
$x\approx ^+_- .59705$
