Find all solutions of the equation in the interval [0, 27). 2 cos 0+1=0 Write your answer in radians in terms of . If there is more than one solution, separate them with commas. 0 = 8 ♫
Find all solutions of the equation in the interval [0, 27). 2 cos 0+1=0 Write your answer in radians in terms of . If there is more than one solution, separate them with commas. 0 = 8 ♫
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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![### Solving Trigonometric Equations
#### Problem Statement
**Find all solutions of the equation in the interval \([0, 2\pi)\).**
\[ 2 \cos \theta + 1 = 0 \]
**Instructions:**
Write your answer in radians in terms of \(\pi\). If there is more than one solution, separate them with commas.
#### Solution Input Box
\[ \theta = \]
---
### Step-by-Step Guide
1. **Understand the Equation:**
The given equation is \( 2 \cos \theta + 1 = 0 \).
2. **Isolate the Trigonometric Function:**
To solve for \(\cos \theta\), subtract 1 from both sides of the equation:
\[ 2 \cos \theta = -1 \]
3. **Solve for \(\cos \theta\):**
Divide both sides by 2:
\[ \cos \theta = -\frac{1}{2} \]
4. **Find the General Solution:**
The cosine function is \(-\frac{1}{2}\) at \( \theta = \frac{2\pi}{3} \) and \( \theta = \frac{4\pi}{3} \) within the interval \([0, 2\pi)\).
5. **Write the Solutions:**
Therefore, the solutions are:
\[ \theta = \frac{2\pi}{3}, \frac{4\pi}{3} \]
6. **Input the Solution:**
Enter the values into the provided answer box, separating them with commas if there is more than one solution:
\[ \theta = \frac{2\pi}{3}, \frac{4\pi}{3} \]
This concludes the example on how to find all solutions for the given trigonometric equation within the specified interval.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F6b574d51-0fdb-495e-9b46-97198cf41098%2F3e8c393f-041a-4d5d-8c92-d229027dec50%2Fvbeoxbl_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Solving Trigonometric Equations
#### Problem Statement
**Find all solutions of the equation in the interval \([0, 2\pi)\).**
\[ 2 \cos \theta + 1 = 0 \]
**Instructions:**
Write your answer in radians in terms of \(\pi\). If there is more than one solution, separate them with commas.
#### Solution Input Box
\[ \theta = \]
---
### Step-by-Step Guide
1. **Understand the Equation:**
The given equation is \( 2 \cos \theta + 1 = 0 \).
2. **Isolate the Trigonometric Function:**
To solve for \(\cos \theta\), subtract 1 from both sides of the equation:
\[ 2 \cos \theta = -1 \]
3. **Solve for \(\cos \theta\):**
Divide both sides by 2:
\[ \cos \theta = -\frac{1}{2} \]
4. **Find the General Solution:**
The cosine function is \(-\frac{1}{2}\) at \( \theta = \frac{2\pi}{3} \) and \( \theta = \frac{4\pi}{3} \) within the interval \([0, 2\pi)\).
5. **Write the Solutions:**
Therefore, the solutions are:
\[ \theta = \frac{2\pi}{3}, \frac{4\pi}{3} \]
6. **Input the Solution:**
Enter the values into the provided answer box, separating them with commas if there is more than one solution:
\[ \theta = \frac{2\pi}{3}, \frac{4\pi}{3} \]
This concludes the example on how to find all solutions for the given trigonometric equation within the specified interval.
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