Answer
$\dfrac{ \pi}{3}$
Work Step by Step
Value of certain composite functions formula:
$f^{-1}(f(x)= \cos^{-1}{(\cos{x})} = x \hspace{25pt} 0 \leq x \leq \pi$
Note that $-\dfrac{5 \pi}{3} \text{ isn't in the interval } \left[0,\pi \right]$.
Recal that:
$\cos{\theta} = \cos{(\theta+2 \pi)}$
Thus,
$\cos{\left(-\dfrac{5 \pi}{3} \right)} = \cos{\left(- \dfrac{5 \pi}{3} +2\pi\right) }= \cos{\left(\dfrac{\pi}{3} \right)}$
Since $\dfrac{\pi}{3} \text{ is in the interval } \left[0,\pi \right]$, then
$\cos^{-1} {\left[\cos{\left(\dfrac{ \pi}{3} \right)} \right]} = \dfrac{ \pi}{3}$
Therefore,
$\cos^{-1} {\left[\cos{\left(-\dfrac{ 5 \pi}{3} \right)} \right]} = \boxed{\dfrac{ \pi}{3}}$
