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.