Write the given expression as an algebraic expression in x. sin(블 cos-1,
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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![**Educational Content: Transforming Trigonometric Expressions**
**Objective:**
Understand how to convert a trigonometric expression into an algebraic expression.
**Problem Statement:**
Write the given expression as an algebraic expression in \( x \).
**Expression:**
\[
\sin\left(\frac{1}{2} \cos^{-1} x\right)
\]
### Explanation
To solve this problem, we are working with the inverse cosine function \(\cos^{-1}(x)\) and the sine of half that angle. We need to express this entirely in terms of \( x \).
1. **Understanding the Inverse Function and Half-Angle:**
- The \(\cos^{-1}(x)\) gives us an angle \(\theta\) such that \(\cos(\theta) = x\).
- We want to find \(\sin\left(\frac{\theta}{2}\right)\).
2. **Apply the Half-Angle Identity for Sine:**
The equation for the sine of a half-angle is:
\[
\sin\left(\frac{\theta}{2}\right) = \sqrt{\frac{1 - \cos(\theta)}{2}}
\]
We know that \(\cos(\theta) = x\), so substituting gives:
\[
\sin\left(\frac{1}{2} \cos^{-1} x\right) = \sqrt{\frac{1 - x}{2}}
\]
### Conclusion
The given trigonometric expression \(\sin\left(\frac{1}{2} \cos^{-1} x\right)\) can be rewritten as the algebraic expression:
\[
\sqrt{\frac{1 - x}{2}}
\]
This expression provides an algebraic representation of the problem in terms of \( x \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Feb07dcd8-0097-4f53-a1a6-006ae808db51%2F1410c27a-91e4-4fec-b8aa-285c24cd4228%2Fgu5kijq_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Educational Content: Transforming Trigonometric Expressions**
**Objective:**
Understand how to convert a trigonometric expression into an algebraic expression.
**Problem Statement:**
Write the given expression as an algebraic expression in \( x \).
**Expression:**
\[
\sin\left(\frac{1}{2} \cos^{-1} x\right)
\]
### Explanation
To solve this problem, we are working with the inverse cosine function \(\cos^{-1}(x)\) and the sine of half that angle. We need to express this entirely in terms of \( x \).
1. **Understanding the Inverse Function and Half-Angle:**
- The \(\cos^{-1}(x)\) gives us an angle \(\theta\) such that \(\cos(\theta) = x\).
- We want to find \(\sin\left(\frac{\theta}{2}\right)\).
2. **Apply the Half-Angle Identity for Sine:**
The equation for the sine of a half-angle is:
\[
\sin\left(\frac{\theta}{2}\right) = \sqrt{\frac{1 - \cos(\theta)}{2}}
\]
We know that \(\cos(\theta) = x\), so substituting gives:
\[
\sin\left(\frac{1}{2} \cos^{-1} x\right) = \sqrt{\frac{1 - x}{2}}
\]
### Conclusion
The given trigonometric expression \(\sin\left(\frac{1}{2} \cos^{-1} x\right)\) can be rewritten as the algebraic expression:
\[
\sqrt{\frac{1 - x}{2}}
\]
This expression provides an algebraic representation of the problem in terms of \( x \).
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