2/3 Your friend is confused with finding the value of sec (). The answer your friend shared with you is . Explain any errors that were made by your friend and then explain how to find the correct answer.
2/3 Your friend is confused with finding the value of sec (). The answer your friend shared with you is . Explain any errors that were made by your friend and then explain how to find the correct answer.
Trigonometry (11th Edition)
11th Edition
ISBN:9780134217437
Author:Margaret L. Lial, John Hornsby, David I. Schneider, Callie Daniels
Publisher:Margaret L. Lial, John Hornsby, David I. Schneider, Callie Daniels
Chapter1: Trigonometric Functions
Section: Chapter Questions
Problem 1RE:
1. Give the measures of the complement and the supplement of an angle measuring 35°.
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![**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 \( \frac{5\pi}{6} \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F487534b5-2eb1-4566-9720-a3993d445361%2F5a381b78-5321-4686-8919-b11055a4fbdb%2F163rksp_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**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 \( \frac{5\pi}{6} \).
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