**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:**\[\sin\left(\frac{1}{2} \cos^{-1} x\right)\]### ExplanationTo 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: \[ \sin\left(\frac{1}{2} \cos^{-1} x\right) = \sqrt{\frac{1 - x}{2}} \]### ConclusionThe 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 \( x \).

Name: **Educational Content: Transforming Trigonometric Expressions****Obje
Uploaded: 2024-03-20T06:46:55.000Z
Channel: Bartleby
Description: **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 \( x \).**E