Solve the multiple-angle equation. (Enter your answers as a comma-separated list. Use n as an integer constant. Enter your response in radians.) 2 sin 4x + V3 = 0
Solve the multiple-angle equation. (Enter your answers as a comma-separated list. Use n as an integer constant. Enter your response in radians.) 2 sin 4x + V3 = 0
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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![**Title: Solving Multiple-Angle Equations**
**Problem Statement:**
Solve the multiple-angle equation. (Enter your answers as a comma-separated list. Use \( n \) as an integer constant. Enter your response in radians.)
\[ 2 \sin (4x) + \sqrt{3} = 0 \]
**Solution:**
This equation can be solved by isolating the variable \( x \).
\[ x = \frac{\pi}{12} + \]
(Here, the given solution seems to be incomplete. Let's go through the steps to complete the solution.)
1. Start by isolating the trigonometric term:
\[ 2 \sin (4x) = -\sqrt{3} \]
2. Divide both sides by 2:
\[ \sin (4x) = -\frac{\sqrt{3}}{2} \]
3. Recall the solutions for \( \sin (y) = -\frac{\sqrt{3}}{2} \). The general solutions are:
\[ y = -\frac{\pi}{3} + 2k\pi \quad \text{and} \quad y = \frac{4\pi}{3} + 2k\pi \quad \text{for integer } k \]
4. Set \( y = 4x \):
\[ 4x = -\frac{\pi}{3} + 2k\pi \quad \text{and} \quad 4x = \frac{4\pi}{3} + 2k\pi \]
5. Divide each equation by 4 to solve for \( x \):
\[ x = -\frac{\pi}{12} + \frac{k\pi}{2} \quad \text{and} \quad x = \frac{\pi}{3} + \frac{k\pi}{2} \]
Thus, the solutions can be written as:
\[ x = -\frac{\pi}{12} + \frac{k\pi}{2}, \quad x = \frac{\pi}{3} + \frac{k\pi}{2} \quad \text{for integer } k \]
**Final Answer:**
\[ x = \frac{\pi}{12} + n\pi \]
Here, \( n \) represents any integer which provides all possible solutions for the given equation in radians.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F5e326e69-8929-40ad-94ce-3b7766681b63%2Fe80bca0d-ffad-4de3-8a19-8c670a74e135%2F761nv79_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Title: Solving Multiple-Angle Equations**
**Problem Statement:**
Solve the multiple-angle equation. (Enter your answers as a comma-separated list. Use \( n \) as an integer constant. Enter your response in radians.)
\[ 2 \sin (4x) + \sqrt{3} = 0 \]
**Solution:**
This equation can be solved by isolating the variable \( x \).
\[ x = \frac{\pi}{12} + \]
(Here, the given solution seems to be incomplete. Let's go through the steps to complete the solution.)
1. Start by isolating the trigonometric term:
\[ 2 \sin (4x) = -\sqrt{3} \]
2. Divide both sides by 2:
\[ \sin (4x) = -\frac{\sqrt{3}}{2} \]
3. Recall the solutions for \( \sin (y) = -\frac{\sqrt{3}}{2} \). The general solutions are:
\[ y = -\frac{\pi}{3} + 2k\pi \quad \text{and} \quad y = \frac{4\pi}{3} + 2k\pi \quad \text{for integer } k \]
4. Set \( y = 4x \):
\[ 4x = -\frac{\pi}{3} + 2k\pi \quad \text{and} \quad 4x = \frac{4\pi}{3} + 2k\pi \]
5. Divide each equation by 4 to solve for \( x \):
\[ x = -\frac{\pi}{12} + \frac{k\pi}{2} \quad \text{and} \quad x = \frac{\pi}{3} + \frac{k\pi}{2} \]
Thus, the solutions can be written as:
\[ x = -\frac{\pi}{12} + \frac{k\pi}{2}, \quad x = \frac{\pi}{3} + \frac{k\pi}{2} \quad \text{for integer } k \]
**Final Answer:**
\[ x = \frac{\pi}{12} + n\pi \]
Here, \( n \) represents any integer which provides all possible solutions for the given equation in radians.
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