Question

Evaluate: \(\cos^{-1} \left( \cos \frac{6\pi}{7} \right)\)

• A. \(\frac{6\pi}{7}\)
• B. \(\frac{6\pi}{5}\)
• C. \(\frac{3\pi}{7}\)
• D. \(\frac{6\pi}{7}\)

Hint:

For \(\cos^{-1}(\cos \theta)\), the result is \(\theta\) when \(\theta\) lies within the range \([0, \pi]\).

Answer 1

The Correct Option is A

Step 1: Recall the principal value range of \(\cos^{-1} x\), which is \(0 \leq \theta \leq \pi\). The cosine function has the property that for \(x\) in the domain of the inverse cosine, \(\cos^{-1}(\cos \theta) = \theta\) when \(\theta\) lies within the range \([0, \pi]\). Step 2: Since \(\frac{6\pi}{7}\) lies within the range \([0, \pi]\), we directly get: \[ \cos^{-1} \left( \cos \frac{6\pi}{7} \right) = \frac{6\pi}{7} \] ---