For the expression below, write an equivalent algebraic expression that involves x only. (Assume x is positive.) sec(cos-15)
For the expression below, write an equivalent algebraic expression that involves x only. (Assume x is positive.) sec(cos-15)
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°.
Related questions
Question
For the expression below, write an equivalent algebraic expression that involves x only. (Assume x is positive.)
sec(cos−1(5/x))
![### Simplifying Trigonometric Expressions
**Problem Statement:**
For the expression below, write an equivalent algebraic expression that involves \( x \) only. (Assume \( x \) is positive):
\[ \sec \left( \cos^{-1} \left( \frac{5}{x} \right) \right) \]
### Steps to Solve:
Let's break this down and find an equivalent expression:
1. **Basic Identities:**
- Recall that \(\cos^{-1} z\) gives an angle \(\theta\) such that \(\cos \theta = z\).
- The \(\sec\) function is the reciprocal of the \(\cos\) function: \(\sec \theta = \frac{1}{\cos \theta}\).
2. **Substitution for Clarity:**
- Let \(\theta = \cos^{-1} \left( \frac{5}{x} \right)\).
- Therefore, \(\cos \theta = \frac{5}{x}\).
3. **Simplification:**
- Using the identity for \(\sec\):
\[ \sec \theta = \frac{1}{\cos \theta} \]
4. **Apply the Identity:**
- Substitute the value of \(\cos \theta\):
\[ \sec \theta = \frac{1}{\frac{5}{x}} = \frac{x}{5} \]
5. **Final Expression:**
- Hence, the expression simplifies to:
\[ \sec \left( \cos^{-1} \left( \frac{5}{x} \right) \right) = \frac{x}{5} \]
**Conclusion:**
Thus, the equivalent algebraic expression for \( \sec \left( \cos^{-1} \left( \frac{5}{x} \right) \right) \) is \( \frac{x}{5} \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa0f44046-e862-45e6-b4db-83887381c0ea%2F6f716335-3042-41e1-9137-4925fd895ec1%2Fayrik5r_processed.png&w=3840&q=75)
Transcribed Image Text:### Simplifying Trigonometric Expressions
**Problem Statement:**
For the expression below, write an equivalent algebraic expression that involves \( x \) only. (Assume \( x \) is positive):
\[ \sec \left( \cos^{-1} \left( \frac{5}{x} \right) \right) \]
### Steps to Solve:
Let's break this down and find an equivalent expression:
1. **Basic Identities:**
- Recall that \(\cos^{-1} z\) gives an angle \(\theta\) such that \(\cos \theta = z\).
- The \(\sec\) function is the reciprocal of the \(\cos\) function: \(\sec \theta = \frac{1}{\cos \theta}\).
2. **Substitution for Clarity:**
- Let \(\theta = \cos^{-1} \left( \frac{5}{x} \right)\).
- Therefore, \(\cos \theta = \frac{5}{x}\).
3. **Simplification:**
- Using the identity for \(\sec\):
\[ \sec \theta = \frac{1}{\cos \theta} \]
4. **Apply the Identity:**
- Substitute the value of \(\cos \theta\):
\[ \sec \theta = \frac{1}{\frac{5}{x}} = \frac{x}{5} \]
5. **Final Expression:**
- Hence, the expression simplifies to:
\[ \sec \left( \cos^{-1} \left( \frac{5}{x} \right) \right) = \frac{x}{5} \]
**Conclusion:**
Thus, the equivalent algebraic expression for \( \sec \left( \cos^{-1} \left( \frac{5}{x} \right) \right) \) is \( \frac{x}{5} \).
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