Problem
Find the exact value, if any, of the composite function. If there is no value, say it is "not defined. "Do not use a calculator. \[ \cos ^{-1}\left[\cos \left(-\frac{23 \pi}{12}\right)\right] \] Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. $\cos ^{-1}\left[\cos \left(-\frac{23 \pi}{12}\right)\right]=\square$ (Simplify your answer. Type an exact answer, using $\pi$ as needed. Use integers or fractions for any numbers in the expression.) B. It is not defined.
Solution
Step 1 ：First, we need to find an equivalent angle to $-\frac{23\pi}{12}$ that lies within the interval $[0, \pi]$. We can do this by adding multiples of $2\pi$ to the angle until it falls within this interval.
Step 2 ：The equivalent angle to $-\frac{23\pi}{12}$ that lies within the interval $[0, \pi]$ is approximately 0.2618. This is the value of the composite function $\cos^{-1}\left[\cos \left(-\frac{23 \pi}{12}\right)\right]$.
Step 3 ：Final Answer: $\cos ^{-1}\left[\cos \left(-\frac{23 \pi}{12}\right)\right]=\boxed{0.2618}$
