$\cos ^{-1}\left[\cos \left(\frac{6 \pi}{7}\right)\right]$
Solution
Step 1 ：The problem is asking for the inverse cosine of the cosine of \(\frac{6 \pi}{7}\). The inverse cosine function, or arccosine, is the function that takes a number in the range of -1 to 1 (the range of the cosine function) and returns the angle in radians whose cosine is that number. The range of the arccosine function is \([0, \pi]\).
Step 2 ：The cosine function is periodic with period \(2\pi\), so \(\cos(\frac{6 \pi}{7})\) is the same as \(\cos(\frac{6 \pi}{7} - 2\pi)\), \(\cos(\frac{6 \pi}{7} - 4\pi)\), and so on.
Step 3 ：However, since the range of the arccosine function is \([0, \pi]\), we need to find a value equivalent to \(\frac{6 \pi}{7}\) that falls within this range. We can do this by subtracting multiples of \(2\pi\) from \(\frac{6 \pi}{7}\) until we get a value within the range of the arccosine function.
Step 4 ：The value of \(\frac{6 \pi}{7}\) after subtracting multiples of \(2\pi\) until it falls within the range \([0, \pi]\) is approximately 2.69. This is the value that the arccosine function will return when given \(\cos(\frac{6 \pi}{7})\) as input.
Step 5 ：Final Answer: \(\cos ^{-1}\left[\cos \left(\frac{6 \pi}{7}\right)\right] = \boxed{2.69}\) (approximately)
