Find the exact value of the expression, if possible. Do not use a calculator. cos^-1[cos(-(pi )/(6))] (5pi )/(6) (pi )/(6) -(pi )/(6) (7pi )/(6)

To find the exact value of the expression $\cos^{-1}[\cos(-\frac{\pi}{6})]$, we need to understand the properties of the inverse cosine function.<br /><br />The inverse cosine function, denoted as $\cos^{-1}(x)$ or $\arccos(x)$, returns the angle whose cosine is equal to $x$. In other words, if $\cos(\theta) = x$, then $\cos^{-1}(x) = \theta$.<br /><br />Now, let's analyze the given expression:<br /><br />$\cos^{-1}[\cos(-\frac{\pi}{6})]$<br /><br />We know that the cosine function is an even function, which means that $\cos(-x) = \cos(x)$. Therefore, we can rewrite the expression as:<br /><br />$\cos^{-1}[\cos(\frac{\pi}{6})]$<br /><br />Since $\cos(\frac{\pi}{6})$ is a positive value, the inverse cosine function will return the angle whose cosine is equal to $\cos(\frac{\pi}{6})$. This angle is $\frac{\pi}{6}$.<br /><br />Therefore, the exact value of the expression $\cos^{-1}[\cos(-\frac{\pi}{6})]$ is $\frac{\pi}{6}$.<br /><br />So, the correct answer is $\frac{\pi}{6}$.