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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![**Trigonometric Equation Solution Guide**
**Problem Statement:**
Solve the trigonometric equation for all values \( 0 \leq x < 2\pi \).
\[ 2 \sin^2 x - 1 = 0 \]
**Instructions:**
1. Isolate the trigonometric function by starting with the equation:
\[ 2 \sin^2 x - 1 = 0 \]
2. Add 1 to both sides to simplify:
\[ 2 \sin^2 x = 1 \]
3. Divide both sides by 2:
\[ \sin^2 x = \frac{1}{2} \]
4. Take the square root of both sides:
\[ \sin x = \pm \frac{\sqrt{2}}{2} \]
5. Find the values of \( x \) within the specified interval \( 0 \leq x < 2\pi \):
- \( \sin x = \frac{\sqrt{2}}{2} \) occurs at \( x = \frac{\pi}{4} \) and \( x = \frac{3\pi}{4} \).
- \( \sin x = -\frac{\sqrt{2}}{2} \) occurs at \( x = \frac{5\pi}{4} \) and \( x = \frac{7\pi}{4} \).
**Solutions:**
- The solutions for \( x \) within the interval \( 0 \leq x < 2\pi \) are:
\[ x = \frac{\pi}{4}, \frac{3\pi}{4}, \frac{5\pi}{4}, \frac{7\pi}{4} \]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc72a8e3d-7815-4c6c-abcb-bdd6024b22ee%2F3844d3ce-a293-4a6c-a81b-bc4783c891fe%2Fsmg8vd06g_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Trigonometric Equation Solution Guide**
**Problem Statement:**
Solve the trigonometric equation for all values \( 0 \leq x < 2\pi \).
\[ 2 \sin^2 x - 1 = 0 \]
**Instructions:**
1. Isolate the trigonometric function by starting with the equation:
\[ 2 \sin^2 x - 1 = 0 \]
2. Add 1 to both sides to simplify:
\[ 2 \sin^2 x = 1 \]
3. Divide both sides by 2:
\[ \sin^2 x = \frac{1}{2} \]
4. Take the square root of both sides:
\[ \sin x = \pm \frac{\sqrt{2}}{2} \]
5. Find the values of \( x \) within the specified interval \( 0 \leq x < 2\pi \):
- \( \sin x = \frac{\sqrt{2}}{2} \) occurs at \( x = \frac{\pi}{4} \) and \( x = \frac{3\pi}{4} \).
- \( \sin x = -\frac{\sqrt{2}}{2} \) occurs at \( x = \frac{5\pi}{4} \) and \( x = \frac{7\pi}{4} \).
**Solutions:**
- The solutions for \( x \) within the interval \( 0 \leq x < 2\pi \) are:
\[ x = \frac{\pi}{4}, \frac{3\pi}{4}, \frac{5\pi}{4}, \frac{7\pi}{4} \]
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