**Trigonometric Equation Problem** **Objective:** Solve the trigonometric equation and provide all solutions. **Equation:** \[ \cos(2x) - \frac{\sqrt{3}}{3} = 0 \] **Instructions:** 1. Isolate the trigonometric function. 2. Identify the values of \(x\) that satisfy the equation within one period of the cosine function. 3. Use the general solution for cosine to find all possible solutions for \(x\). **Solution Approach:** 1. Start by adding \(\frac{\sqrt{3}}{3}\) to both sides of the equation to get: \[ \cos(2x) = \frac{\sqrt{3}}{3} \] 2. Identify angles \(2x\) that have a cosine of \(\frac{\sqrt{3}}{3}\). 3. Write the general solution for \(x\): \[ x = \frac{1}{2}(\cos^{-1}\left(\frac{\sqrt{3}}{3}\right) + 2n\pi) \quad \text{and} \quad x = \frac{1}{2}(-\cos^{-1}\left(\frac{\sqrt{3}}{3}\right) + 2m\pi) \] where \(n, m \in \mathbb{Z}\). Explore the properties of the cosine function and verify if additional solutions exist within the required domain.
**Trigonometric Equation Problem** **Objective:** Solve the trigonometric equation and provide all solutions. **Equation:** \[ \cos(2x) - \frac{\sqrt{3}}{3} = 0 \] **Instructions:** 1. Isolate the trigonometric function. 2. Identify the values of \(x\) that satisfy the equation within one period of the cosine function. 3. Use the general solution for cosine to find all possible solutions for \(x\). **Solution Approach:** 1. Start by adding \(\frac{\sqrt{3}}{3}\) to both sides of the equation to get: \[ \cos(2x) = \frac{\sqrt{3}}{3} \] 2. Identify angles \(2x\) that have a cosine of \(\frac{\sqrt{3}}{3}\). 3. Write the general solution for \(x\): \[ x = \frac{1}{2}(\cos^{-1}\left(\frac{\sqrt{3}}{3}\right) + 2n\pi) \quad \text{and} \quad x = \frac{1}{2}(-\cos^{-1}\left(\frac{\sqrt{3}}{3}\right) + 2m\pi) \] where \(n, m \in \mathbb{Z}\). Explore the properties of the cosine function and verify if additional solutions exist within the required domain.
Trigonometry (11th Edition)
11th Edition
ISBN:9780134217437
Author:Margaret L. Lial, John Hornsby, David I. Schneider, Callie Daniels
Publisher:Margaret L. Lial, John Hornsby, David I. Schneider, Callie Daniels
Chapter1: Trigonometric Functions
Section: Chapter Questions
Problem 1RE:
1. Give the measures of the complement and the supplement of an angle measuring 35°.
Related questions
Question
Give all solutions
![**Trigonometric Equation Problem**
**Objective:** Solve the trigonometric equation and provide all solutions.
**Equation:**
\[
\cos(2x) - \frac{\sqrt{3}}{3} = 0
\]
**Instructions:**
1. Isolate the trigonometric function.
2. Identify the values of \(x\) that satisfy the equation within one period of the cosine function.
3. Use the general solution for cosine to find all possible solutions for \(x\).
**Solution Approach:**
1. Start by adding \(\frac{\sqrt{3}}{3}\) to both sides of the equation to get:
\[
\cos(2x) = \frac{\sqrt{3}}{3}
\]
2. Identify angles \(2x\) that have a cosine of \(\frac{\sqrt{3}}{3}\).
3. Write the general solution for \(x\):
\[
x = \frac{1}{2}(\cos^{-1}\left(\frac{\sqrt{3}}{3}\right) + 2n\pi) \quad \text{and} \quad x = \frac{1}{2}(-\cos^{-1}\left(\frac{\sqrt{3}}{3}\right) + 2m\pi)
\]
where \(n, m \in \mathbb{Z}\).
Explore the properties of the cosine function and verify if additional solutions exist within the required domain.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3e8ca5a6-77b3-4f88-a5f3-282372d2c0e0%2Fa0f95a18-acb5-4e63-a624-c050b84bd367%2Fhj53g9_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Trigonometric Equation Problem**
**Objective:** Solve the trigonometric equation and provide all solutions.
**Equation:**
\[
\cos(2x) - \frac{\sqrt{3}}{3} = 0
\]
**Instructions:**
1. Isolate the trigonometric function.
2. Identify the values of \(x\) that satisfy the equation within one period of the cosine function.
3. Use the general solution for cosine to find all possible solutions for \(x\).
**Solution Approach:**
1. Start by adding \(\frac{\sqrt{3}}{3}\) to both sides of the equation to get:
\[
\cos(2x) = \frac{\sqrt{3}}{3}
\]
2. Identify angles \(2x\) that have a cosine of \(\frac{\sqrt{3}}{3}\).
3. Write the general solution for \(x\):
\[
x = \frac{1}{2}(\cos^{-1}\left(\frac{\sqrt{3}}{3}\right) + 2n\pi) \quad \text{and} \quad x = \frac{1}{2}(-\cos^{-1}\left(\frac{\sqrt{3}}{3}\right) + 2m\pi)
\]
where \(n, m \in \mathbb{Z}\).
Explore the properties of the cosine function and verify if additional solutions exist within the required domain.
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