3. Find all solutions: 2cos (2x -) + (2x -") 3 = 4 in the interval [0,2n) %3D 6.
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°.
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![**Problem:**
Find all solutions for the equation \( 2\cos\left(2x - \frac{\pi}{6}\right) + 3 = 4 \) within the interval \([0, 2\pi)\).
**Solution Approach:**
1. **Rearrange the Equation:**
Start by isolating the cosine function:
\[
2\cos\left(2x - \frac{\pi}{6}\right) = 1
\]
Divide by 2:
\[
\cos\left(2x - \frac{\pi}{6}\right) = \frac{1}{2}
\]
2. **Solve for the Angle:**
The cosine of an angle is \(\frac{1}{2}\) at specific standard angles. Within one full rotation (0 to \(2\pi\)), \(\cos(\theta) = \frac{1}{2}\) when:
\[
\theta = \frac{\pi}{3} \quad \text{or} \quad \theta = \frac{5\pi}{3}
\]
3. **Express in Terms of \(x\):**
Set \(2x - \frac{\pi}{6}\) equal to each of the angles found:
- \( 2x - \frac{\pi}{6} = \frac{\pi}{3} \)
- \( 2x - \frac{\pi}{6} = \frac{5\pi}{3} \)
Solve each equation for \(x\):
- For \(2x - \frac{\pi}{6} = \frac{\pi}{3} \):
\[
2x = \frac{\pi}{3} + \frac{\pi}{6} = \frac{2\pi}{6} + \frac{\pi}{6} = \frac{3\pi}{6} = \frac{\pi}{2}
\]
\[
x = \frac{\pi}{4}
\]
- For \(2x - \frac{\pi}{6} = \frac{5\pi}{3} \):
\[
2x = \frac{5\pi}{3} + \frac{\pi}{6} = \frac{10\pi}{6} + \frac](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fcf75519a-4b73-4600-b1a2-6168f4806ba2%2Fa6e1687e-0c88-44c8-ae1b-662705a48e2b%2Ftw4g9te.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem:**
Find all solutions for the equation \( 2\cos\left(2x - \frac{\pi}{6}\right) + 3 = 4 \) within the interval \([0, 2\pi)\).
**Solution Approach:**
1. **Rearrange the Equation:**
Start by isolating the cosine function:
\[
2\cos\left(2x - \frac{\pi}{6}\right) = 1
\]
Divide by 2:
\[
\cos\left(2x - \frac{\pi}{6}\right) = \frac{1}{2}
\]
2. **Solve for the Angle:**
The cosine of an angle is \(\frac{1}{2}\) at specific standard angles. Within one full rotation (0 to \(2\pi\)), \(\cos(\theta) = \frac{1}{2}\) when:
\[
\theta = \frac{\pi}{3} \quad \text{or} \quad \theta = \frac{5\pi}{3}
\]
3. **Express in Terms of \(x\):**
Set \(2x - \frac{\pi}{6}\) equal to each of the angles found:
- \( 2x - \frac{\pi}{6} = \frac{\pi}{3} \)
- \( 2x - \frac{\pi}{6} = \frac{5\pi}{3} \)
Solve each equation for \(x\):
- For \(2x - \frac{\pi}{6} = \frac{\pi}{3} \):
\[
2x = \frac{\pi}{3} + \frac{\pi}{6} = \frac{2\pi}{6} + \frac{\pi}{6} = \frac{3\pi}{6} = \frac{\pi}{2}
\]
\[
x = \frac{\pi}{4}
\]
- For \(2x - \frac{\pi}{6} = \frac{5\pi}{3} \):
\[
2x = \frac{5\pi}{3} + \frac{\pi}{6} = \frac{10\pi}{6} + \frac
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