Find all of the solutions of the equation on the interval [0, 2+]: secxccx=2cscx. O 풍, 풍 O 89°F Partly sunny O $7,0, n, 2n 1117 0, n, 2n Q Search J
Find all of the solutions of the equation on the interval [0, 2+]: secxccx=2cscx. O 풍, 풍 O 89°F Partly sunny O $7,0, n, 2n 1117 0, n, 2n Q Search J
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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Question
![**Title:** Solving Trigonometric Equations on a Given Interval
**Problem Statement:**
Find all of the solutions of the equation on the interval \([0, 2\pi]\): \(\sec x \csc x = 2 \csc x\).
**Answer Choices:**
1. \(\frac{\pi}{3}, \frac{5\pi}{3}\)
2. \(\frac{\pi}{3}, \frac{5\pi}{3}, 0, \pi, 2\pi\)
3. \(\frac{\pi}{6}, \frac{11\pi}{6}\)
4. \(0, \pi, 2\pi\)
**Explanation:**
The question involves finding the solutions for the given trigonometric equation within a specified interval. Let's break down the process to solve the equation:
1. **Rewrite the Equation:**
\[ \sec x \csc x = 2 \csc x \]
2. **Simplify the Equation:**
- Since \(\csc x \neq 0\) in the given interval, divide both sides of the equation by \(\csc x\):
\[ \sec x = 2 \]
3. **Convert to Cosine:**
- Recall that \(\sec x = \frac{1}{\cos x}\):
\[ \frac{1}{\cos x} = 2 \]
4. **Solve for \(\cos x\):**
\[ \cos x = \frac{1}{2} \]
5. **Find \(x\) in the Interval \([0, 2\pi]\):**
- The cosine function \(\cos x = \frac{1}{2}\) at:
\[ x = \frac{\pi}{3}, \frac{5\pi}{3} \]
Therefore, the solutions for the equation \(\sec x \csc x = 2 \csc x\) in the interval \([0, 2\pi]\) are:
\[ x = \frac{\pi}{3}, \frac{5\pi}{3} \]
The correct answer is:
1. \(\frac{\pi}{3}, \frac{5\pi}{3}\)
---
This content explains the problem, outlines the process of solving the trigonometric equation step-by-step, and identifies the correct solution set.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F6f02248d-9b04-4bf9-ba6b-18c74681389d%2Ff97bd670-1ab5-453a-b707-de0bf8e70394%2Fnf4l604_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Title:** Solving Trigonometric Equations on a Given Interval
**Problem Statement:**
Find all of the solutions of the equation on the interval \([0, 2\pi]\): \(\sec x \csc x = 2 \csc x\).
**Answer Choices:**
1. \(\frac{\pi}{3}, \frac{5\pi}{3}\)
2. \(\frac{\pi}{3}, \frac{5\pi}{3}, 0, \pi, 2\pi\)
3. \(\frac{\pi}{6}, \frac{11\pi}{6}\)
4. \(0, \pi, 2\pi\)
**Explanation:**
The question involves finding the solutions for the given trigonometric equation within a specified interval. Let's break down the process to solve the equation:
1. **Rewrite the Equation:**
\[ \sec x \csc x = 2 \csc x \]
2. **Simplify the Equation:**
- Since \(\csc x \neq 0\) in the given interval, divide both sides of the equation by \(\csc x\):
\[ \sec x = 2 \]
3. **Convert to Cosine:**
- Recall that \(\sec x = \frac{1}{\cos x}\):
\[ \frac{1}{\cos x} = 2 \]
4. **Solve for \(\cos x\):**
\[ \cos x = \frac{1}{2} \]
5. **Find \(x\) in the Interval \([0, 2\pi]\):**
- The cosine function \(\cos x = \frac{1}{2}\) at:
\[ x = \frac{\pi}{3}, \frac{5\pi}{3} \]
Therefore, the solutions for the equation \(\sec x \csc x = 2 \csc x\) in the interval \([0, 2\pi]\) are:
\[ x = \frac{\pi}{3}, \frac{5\pi}{3} \]
The correct answer is:
1. \(\frac{\pi}{3}, \frac{5\pi}{3}\)
---
This content explains the problem, outlines the process of solving the trigonometric equation step-by-step, and identifies the correct solution set.
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