To solve the equation \((-2) \cos(2x + \frac{\pi}{4}) = 1\), follow these steps:

### Step 1: Isolate the cosine function.
Divide both sides of the equation by \(-2\):
\[
\cos\left(2x + \frac{\pi}{4}\right) = -\frac{1}{2}
\]

### Step 2: Determine the angle for which cosine equals \(-\frac{1}{2}\).
The cosine of \(-\frac{1}{2}\) occurs at:
\[
\theta = \frac{2\pi}{3} + 2n\pi \quad \text{and} \quad \theta = \frac{4\pi}{3} + 2n\pi, \quad \text{where } n \text{ is any integer.}
\]
Therefore,
\[
2x + \frac{\pi}{4} = \frac{2\pi}{3} + 2n\pi \quad \text{or} \quad 2x + \frac{\pi}{4} = \frac{4\pi}{3} + 2n\pi
\]

### Step 3: Solve for \(x\).
For the first case:
\[
2x = \frac{2\pi}{3} - \frac{\pi}{4} + 2n\pi
\]
\[
2x = \frac{8\pi - 3\pi}{12} + 2n\pi
\]
\[
2x = \frac{5\pi}{12} + 2n\pi
\]
\[
x = \frac{5\pi}{24} + n\pi
\]

For the second case:
\[
2x = \frac{4\pi}{3} - \frac{\pi}{4} + 2n\pi
\]
\[
2x = \frac{16\pi - 3\pi}{12} + 2n\pi
\]
\[
2x = \frac{13\pi}{12} + 2n\pi
\]
\[
x = \frac{13\pi}{24} + n\pi
\]

### Final Solutions:
The general solutions for \(x\) are:
\[
x = \frac{5\pi}{24} + n\pi \quad \text{or} \quad x = \frac{13\pi}{24} + n\pi, \quad \text{where } n \text{ is any integer.}
\]

Would you like more details, or do you have any questions?

### Related Questions:
1. What is the general form of the cosine function and its transformations?
2. How do you solve trigonometric equations involving multiple angles?
3. What are the key differences between solving cosine and sine equations?
4. How can you determine the exact solutions for sine or cosine equations in different intervals?
5. How does the period of a trigonometric function affect the solutions of an equation?

### Tip:
Always check if your solutions fall within the required domain, especially when solving trigonometric equations.

Learn how to solve the trigonometric equation (-2) cos(2x + pi/4) = 1 step-by-step with detailed solutions. This problem involves trigonometric identities and solving equations with multiple angles. Suitable for advanced high school students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation