To solve the equation \( \cos(2x) = 0 \), follow these steps:

### Step 1: Understand the basic trigonometric identity
The equation \( \cos(2x) = 0 \) means we're looking for values of \( 2x \) where the cosine function equals zero.

### Step 2: Determine the general solution for cosine equal to zero
The cosine function equals zero at:
\[ \cos\left(\frac{\pi}{2} + n\pi\right) = 0 \]
where \( n \) is an integer.

### Step 3: Solve for \( 2x \)
Set \( 2x \) equal to the angles where the cosine is zero:
\[ 2x = \frac{\pi}{2} + n\pi \]
\[ 2x = \frac{\pi}{2} + n\pi \quad \text{where } n \in \mathbb{Z} \]

### Step 4: Solve for \( x \)
Divide both sides by 2:
\[ x = \frac{\pi}{4} + \frac{n\pi}{2} \quad \text{where } n \in \mathbb{Z} \]

### Final Answer
The general solution for the equation \( \cos(2x) = 0 \) is:
\[ x = \frac{\pi}{4} + \frac{n\pi}{2} \quad \text{where } n \in \mathbb{Z} \]

This solution represents an infinite number of solutions based on the integer \( n \).

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Would you like more details on this solution or have any other questions?

Here are 5 related questions you might find interesting:

1. How do you solve \( \cos(3x) = 0 \)?
2. What is the general solution for \( \sin(2x) = 0 \)?
3. How does the solution to \( \cos(x) = 0 \) differ from \( \cos(2x) = 0 \)?
4. How do you find the exact values for \( x \) in a specific interval for \( \cos(2x) = 0 \)?
5. Can you explain how the period of the cosine function affects the solution to \( \cos(2x) = 0 \)?

**Tip:** Remember that the general solution for trigonometric equations often involves adding a multiple of the period to account for all possible solutions.

Learn how to solve the trigonometric equation cos(2x) = 0 with a detailed solution. This guide explains key trigonometric concepts, including the cosine function, general solutions for trigonometric equations, and the periodicity of the cosine function. Suitable for students in Grades 9-12.

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