Question

The value of \( \cos^{-1} \left( \cos \frac{7\pi}{6} \right) \) is

• A. \( \frac{7\pi}{6} \)
• B. \( \frac{5\pi}{3} \)
• C. \( \frac{5\pi}{6} \)
• D. \( \frac{13\pi}{6} \)

Hint:

For inverse trigonometric functions, always consider the range of the function and use reference angles to find equivalent values.

Answer 1

The Correct Option is C

Step 1: Recall the range of the inverse cosine function.
The range of the inverse cosine function \( \cos^{-1}(x) \) is \( [0, \pi] \), meaning that the output of the inverse cosine is always within this interval.

Step 2: Analyze the angle \( \frac{7\pi}{6} \).
The angle \( \frac{7\pi}{6} \) lies in the third quadrant. To find the corresponding angle in the range \( [0, \pi] \), we need to use the reference angle.

Step 3: Find the reference angle.
The reference angle for \( \frac{7\pi}{6} \) is: \[ \frac{7\pi}{6} - \pi = \frac{\pi}{6} \]

Step 4: Apply the inverse cosine.
Since \( \cos(\frac{7\pi}{6}) = \cos(\frac{\pi}{6}) \), and the inverse cosine function returns values in the range \( [0, \pi] \), we have: \[ \cos^{-1} \left( \cos \frac{7\pi}{6} \right) = \frac{5\pi}{6} \]

Step 5: Conclusion.
Thus, the value of \( \cos^{-1} \left( \cos \frac{7\pi}{6} \right) \) is \( \frac{5\pi}{6} \), corresponding to option (C).