find the exact value of the expression, if it is defined. (if an answer is undefined, enter…

Explanation:

Step1: Rewrite the angle

We know that $\frac{11\pi}{6}=2\pi-\frac{\pi}{6}$. So, $\cos(\frac{11\pi}{6})=\cos(2\pi - \frac{\pi}{6})$. Since $\cos(2k\pi - \alpha)=\cos\alpha$ for integer $k$, here $k = 1$, then $\cos(2\pi-\frac{\pi}{6})=\cos(\frac{\pi}{6})$.

Step2: Use inverse - cosine property

The function $y = \cos^{-1}(x)$ has a domain of $[- 1,1]$ and range of $[0,\pi]$. We need to find $\cos^{-1}(\cos(\frac{11\pi}{6}))$, and since $\cos(\frac{11\pi}{6})=\cos(\frac{\pi}{6})$, then $\cos^{-1}(\cos(\frac{11\pi}{6}))=\cos^{-1}(\cos(\frac{\pi}{6}))$. For $x\in[0,\pi]$, $\cos^{-1}(\cos(x)) = x$. And $\frac{\pi}{6}\in[0,\pi]$, so $\cos^{-1}(\cos(\frac{11\pi}{6}))=\frac{\pi}{6}$.

Answer:

$\frac{\pi}{6}$