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
![**Trigonometric Equation Solutions**
**Objective:**
Find all of the solutions of the given trigonometric equation on the interval \([0, 2\pi]\):
\[ \sec{x} \csc{x} = 2 \csc{x} \]
**Options:**
1. \(\frac{\pi}{3}, \frac{5\pi}{3}, 0, \pi, 2\pi\)
2. \(\frac{\pi}{6}, \frac{11\pi}{6}\)
3. \(\frac{\pi}{3}, \frac{5\pi}{3}\)
4. \(0, \pi, 2\pi\)
**Explanation of the Options:**
1. **Option 1:**
This option includes the angles \(\frac{\pi}{3}\), \(\frac{5\pi}{3}\), \(0\), \(\pi\), and \(2\pi\).
2. **Option 2:**
This option includes the angles \(\frac{\pi}{6}\) and \(\frac{11\pi}{6}\).
3. **Option 3:**
This option includes the angles \(\frac{\pi}{3}\) and \(\frac{5\pi}{3}\).
4. **Option 4:**
This option includes the angles \(0\), \(\pi\), and \(2\pi\).
**Instructions:**
1. Simplify and solve the given equation.
2. Verify the solutions fall within the interval \([0, 2\pi]\).
3. Compare the derived solutions to the provided options.
To understand and solve the equation comprehensively, utilize trigonometric identities and their properties, ensuring a systematic verification of each solution in the specified interval.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fcccb93a5-bda7-4c84-b750-51d9219067a4%2F2982a458-d5c2-44d7-b4d9-8c791f02d6f2%2Fsyvnpj_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Trigonometric Equation Solutions**
**Objective:**
Find all of the solutions of the given trigonometric equation on the interval \([0, 2\pi]\):
\[ \sec{x} \csc{x} = 2 \csc{x} \]
**Options:**
1. \(\frac{\pi}{3}, \frac{5\pi}{3}, 0, \pi, 2\pi\)
2. \(\frac{\pi}{6}, \frac{11\pi}{6}\)
3. \(\frac{\pi}{3}, \frac{5\pi}{3}\)
4. \(0, \pi, 2\pi\)
**Explanation of the Options:**
1. **Option 1:**
This option includes the angles \(\frac{\pi}{3}\), \(\frac{5\pi}{3}\), \(0\), \(\pi\), and \(2\pi\).
2. **Option 2:**
This option includes the angles \(\frac{\pi}{6}\) and \(\frac{11\pi}{6}\).
3. **Option 3:**
This option includes the angles \(\frac{\pi}{3}\) and \(\frac{5\pi}{3}\).
4. **Option 4:**
This option includes the angles \(0\), \(\pi\), and \(2\pi\).
**Instructions:**
1. Simplify and solve the given equation.
2. Verify the solutions fall within the interval \([0, 2\pi]\).
3. Compare the derived solutions to the provided options.
To understand and solve the equation comprehensively, utilize trigonometric identities and their properties, ensuring a systematic verification of each solution in the specified interval.
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