**Problem Explanation:**Your friend is confused with finding the value of \( \sec^{-1}\left(-\frac{2\sqrt{3}}{3}\right) \). The answer your friend shared with you is \( -\frac{\pi}{6} \).**Task:**1. Explain any errors that were made by your friend.2. Explain how to find the correct answer.**Analysis and Correct Solution:**1. **Understanding the Secant Inverse Function:** - The secant inverse function, \( \sec^{-1}(x) \), gives the angle \( \theta \) such that \( \sec(\theta) = x \). - The range of \( \sec^{-1}(x) \) is \([0, \pi]\), excluding \( \frac{\pi}{2} \).2. **Evaluating Errors:** - The value \( -\frac{\pi}{6} \) is not in the range \([0, \pi]\). - A negative angle like \(-\frac{\pi}{6}\) cannot be the correct principal value since it is not within the defined range of the secant inverse function.3. **Correct Approach:** - Recognize that \( \sec(\theta) = \frac{1}{\cos(\theta)} \), so we solve \( \frac{1}{\cos(\theta)} = -\frac{2\sqrt{3}}{3} \). - Rearrange to find \( \cos(\theta) = -\frac{\sqrt{3}}{2} \). - Determine the angles where \( \cos(\theta) = -\frac{\sqrt{3}}{2} \) within the range \([0, \pi]\). - This occurs at \( \theta = \frac{5\pi}{6} \).4. **Conclusion:** - The correct value of \( \sec^{-1}\left(-\frac{2\sqrt{3}}{3}\right) \) is \( \frac{5\pi}{6} \).

Name: **Problem Explanation:**Your friend is confused with finding the valu
Uploaded: 2024-03-20T06:46:55.000Z
Channel: Bartleby
Description: **Problem Explanation:**Your friend is confused with finding the value of \( \sec^{-1}\left(-\frac{2\sqrt{3}}{3}\right) \). The answer your friend shared with you is \( -\frac{\pi}{6} \).**Task:**1. Explain any errors that were made by your friend.2