Here's a precalculus problem involving trigonometry and functions:

### Problem:
Given the function \( f(x) = 2\sin(x) + \cos(2x) \), solve for all values of \( x \) in the interval \( [0, 2\pi] \) where \( f(x) = 0 \).

### Steps:
1. Find where \( 2\sin(x) + \cos(2x) = 0 \).
2. Use the identity \( \cos(2x) = 1 - 2\sin^2(x) \) to rewrite the equation in terms of \( \sin(x) \).
3. Solve the resulting equation and find the corresponding values of \( x \) in the interval \( [0, 2\pi] \).

Would you like me to work through the solution step by step?

### Follow-up questions:
1. How do we use trigonometric identities to simplify equations?
2. What is the general solution to \( \sin(x) = 0 \) or \( \cos(x) = 0 \)?
3. How does the double angle identity for cosine work?
4. How do we graph \( 2\sin(x) + \cos(2x) \) to verify solutions?
5. What is the importance of solving trigonometric equations within a specified interval?

### Tip:
When working with trigonometric equations, always check if applying an identity simplifies the problem before proceeding with algebraic steps.

Explore a precalculus problem involving trigonometric identities and functions. Learn how to solve the equation 2sin(x) + cos(2x) = 0 for all values of x in the interval [0, 2π] using trigonometric identities. Suitable for Grades 11-12 students.

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