### Problem Statement:**Rewrite** \( \sin \left( 2 \cos^{-1} \dfrac{u}{4} \right) \) **as an algebraic expression in** \( u \).### Solution:We start with the given expression:\[ \sin \left( 2 \cos^{-1} \dfrac{u}{4} \right) \]### Explanation:1. **Understanding Inverse Cosine**: - \( \cos^{-1}(y) \) returns the angle whose cosine is \( y \). So \( \cos(\cos^{-1}(y)) = y \). 2. **Double Angle Formula for Sine**: - The double angle formula for sine states: \( \sin(2\theta) = 2 \sin(\theta) \cos(\theta) \).3. **Using Substitution**: - Let \( \theta = \cos^{-1} \left( \dfrac{u}{4} \right) \), so \( \cos(\theta) = \dfrac{u}{4} \).4. **Finding** \( \sin(\theta) \): - Using the Pythagorean identity: \( \sin^2(\theta) + \cos^2(\theta) = 1 \), - \( \sin^2(\theta) = 1 - \cos^2(\theta) = 1 - \left( \dfrac{u}{4} \right)^2 \). Thus: \[ \sin(\theta) = \sqrt{1 - \left( \dfrac{u}{4} \right)^2} \]5. **Putting it Together**: - Now use the double angle formula: \[ \sin(2\theta) = 2 \sin(\theta) \cos(\theta) = 2 \left( \sqrt{1 - \left( \dfrac{u}{4} \right)^2} \right) \left( \dfrac{u}{4} \right) = \dfrac{2u}{4} \sqrt{1 - \left( \dfrac{u}{4} \right)^2} = \dfrac{u}{2} \sqrt{4 - u^2} \]### Conclusion:So, the algebra

Name: ### Problem Statement:**Rewrite** \( \sin \left( 2 \cos^{-1} \dfrac{u
Uploaded: 2024-03-20T06:46:55.000Z
Channel: Bartleby
Description: ### Problem Statement:**Rewrite** \( \sin \left( 2 \cos^{-1} \dfrac{u}{4} \right) \) **as an algebraic expression in** \( u \).### Solution:We start with the given expression:\[ \sin \left( 2 \cos^{-1} \dfrac{u}{4} \right) \]### Explanation:1. **Und