https://file.aiquickdraw.com/mathbot/file/17271021035727qwgsrm5.jpg

The equation in the image begins with \( \sqrt{2} \cos(2\pi x - \frac{\pi}{3}) + 1 = \sqrt{2} + 1 \). Let's work through this equation:

1. **Simplifying the equation**:
   \[
   \sqrt{2} \cos\left(2\pi x - \frac{\pi}{3}\right) + 1 = \sqrt{2} + 1
   \]
   Subtract 1 from both sides:
   \[
   \sqrt{2} \cos\left(2\pi x - \frac{\pi}{3}\right) = \sqrt{2}
   \]

2. **Divide by \( \sqrt{2} \)**:
   \[
   \cos\left(2\pi x - \frac{\pi}{3}\right) = 1
   \]

3. **Solving for the angle**:
   We know that \( \cos(\theta) = 1 \) when \( \theta = 2n\pi \) (where \( n \) is an integer). Therefore:
   \[
   2\pi x - \frac{\pi}{3} = 2n\pi
   \]
   Add \( \frac{\pi}{3} \) to both sides:
   \[
   2\pi x = 2n\pi + \frac{\pi}{3}
   \]
   Divide by \( 2\pi \):
   \[
   x = n + \frac{1}{6}
   \]

So the general solution is:
\[
x = n + \frac{1}{6}, \quad n \in \mathbb{Z}
\]

Would you like more details on specific steps or have any other questions?

Here are 5 related questions for further exploration:
1. What other values of cosine function produce the same result?
2. How can the equation be represented graphically?
3. How do phase shifts affect trigonometric equations?
4. What is the general solution for \( \sin \theta = 1 \)?
5. How does solving trigonometric equations differ from polynomial equations?

**Tip:** Always check your solutions within the original equation to ensure all steps were followed correctly!

Solve the trigonometric equation in the image: √2 cos(2πx - π/3) + 1 = √2 + 1

This page explains how to solve the trigonometric equation √2 cos(2πx - π/3) + 1 = √2 + 1 with detailed step-by-step calculations. It covers key trigonometric concepts, including solving for angles using cosine and general solutions for trigonometric equations. Suitable for students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation