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Let's solve the two given trigonometric problems.

### Problem 1: Solve \( \sin(2x) = 1 \)

The general solution for \( \sin(2x) = 1 \) is when:
\[
2x = \frac{\pi}{2} + 2k\pi, \quad k \in \mathbb{Z}
\]

To solve for \( x \), divide by 2:
\[
x = \frac{\pi}{4} + k\pi, \quad k \in \mathbb{Z}
\]

Now let's compare this with the provided options:

a) \( n\pi + (-1)^n \frac{\pi}{2} \)  
b) \( n\pi + (-1)^n \frac{\pi}{4} \)  
c) \( \frac{\pi}{2} + (-1)^n \frac{\pi}{4} \)  
d) \( n\pi + (-1)^n \frac{\pi}{8} \)  
e) \( \frac{\pi}{2} + (-1)^n \pi \)  

The correct answer matches option **b)**: \( n\pi + (-1)^n \frac{\pi}{4} \).

### Problem 2: Solve \( \sin(3x) = 0 \)

The general solution for \( \sin(3x) = 0 \) is when:
\[
3x = k\pi, \quad k \in \mathbb{Z}
\]

To solve for \( x \), divide by 3:
\[
x = \frac{k\pi}{3}, \quad k \in \mathbb{Z}
\]

Now, let's match this with the provided options:

a) \( n\pi \)  
b) \( \frac{n\pi}{3} \)  
c) \( 3n\pi \)  
d) \( n\pi + (-1)^n \frac{\pi}{2} \)  
e) \( \frac{n\pi}{3} + (-1)^n \frac{\pi}{6} \)  

The correct answer matches option **b)**: \( \frac{n\pi}{3} \).

### Summary of Answers
1. The correct answer for Problem 1 is **b)** \( n\pi + (-1)^n \frac{\pi}{4} \).
2. The correct answer for Problem 2 is **b)** \( \frac{n\pi}{3} \).

Would you like more details on the solutions, or do you have any further questions?

### Related Questions
1. How to solve equations involving \( \cos(x) \)?
2. What is the general solution for \( \sin(x) = -1 \)?
3. How to solve trigonometric equations with multiple angles, like \( \sin(4x) \)?
4. What is the process for solving equations involving both \( \sin(x) \) and \( \cos(x) \)?
5. How can trigonometric identities help solve equations?

**Tip:** Always remember to express the general solution in terms of \( n \) or \( k \) (integers) when solving trigonometric equations.

Solve the following trigonometric equations: sen(2x) = 1 and sen(3x) = 0.

Learn how to solve the trigonometric equations sin(2x) = 1 and sin(3x) = 0, with detailed solutions. Explore the key trigonometric concepts, such as the sine function and general solutions for trigonometric equations, suitable for students in Grades 10-12.

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