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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![### Problem Statement:
14. Solve \(4 \sin^2 x - 1 = 0\) for principal values of \(x\). Express the solution(s) in degrees.
### Solution:
To solve the equation \(4 \sin^2 x - 1 = 0\), follow these steps:
1. **Simplify the Equation:**
\[
4 \sin^2 x - 1 = 0
\]
Add 1 to both sides:
\[
4 \sin^2 x = 1
\]
Divide both sides by 4:
\[
\sin^2 x = \frac{1}{4}
\]
2. **Solve for \(\sin x\):**
Take the square root of both sides:
\[
\sin x = \pm \frac{1}{2}
\]
3. **Find the Principal Values for \(x\):**
- For \(\sin x = \frac{1}{2}\), the principal values are:
\[
x = 30^\circ \quad \text{and} \quad x = 150^\circ
\]
- For \(\sin x = -\frac{1}{2}\), the principal values are:
\[
x = 210^\circ \quad \text{and} \quad x = 330^\circ
\]
### Conclusion:
The solutions for \(x\) in degrees are:
\[
x = 30^\circ, \; 150^\circ, \; 210^\circ, \; 330^\circ
\]
This solution approach verifies that \(x\) values satisfy \(4 \sin^2 x - 1 = 0\).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb805b29b-1703-42be-83a9-c544ba7b8f1b%2Ff1c74b98-af99-44c3-8ee6-8d893855ea69%2Fimaem1_processed.png&w=3840&q=75)
Transcribed Image Text:### Problem Statement:
14. Solve \(4 \sin^2 x - 1 = 0\) for principal values of \(x\). Express the solution(s) in degrees.
### Solution:
To solve the equation \(4 \sin^2 x - 1 = 0\), follow these steps:
1. **Simplify the Equation:**
\[
4 \sin^2 x - 1 = 0
\]
Add 1 to both sides:
\[
4 \sin^2 x = 1
\]
Divide both sides by 4:
\[
\sin^2 x = \frac{1}{4}
\]
2. **Solve for \(\sin x\):**
Take the square root of both sides:
\[
\sin x = \pm \frac{1}{2}
\]
3. **Find the Principal Values for \(x\):**
- For \(\sin x = \frac{1}{2}\), the principal values are:
\[
x = 30^\circ \quad \text{and} \quad x = 150^\circ
\]
- For \(\sin x = -\frac{1}{2}\), the principal values are:
\[
x = 210^\circ \quad \text{and} \quad x = 330^\circ
\]
### Conclusion:
The solutions for \(x\) in degrees are:
\[
x = 30^\circ, \; 150^\circ, \; 210^\circ, \; 330^\circ
\]
This solution approach verifies that \(x\) values satisfy \(4 \sin^2 x - 1 = 0\).
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