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The given equation is:

\[
2 \sin(x) + \sin(2x) = 0
\]

We are tasked with finding all solutions for \( x \) in the interval \([0, 2\pi]\), and the answers should be in radians in terms of \(\pi\).

### Step-by-step solution:

1. **Use the double angle identity** for \(\sin(2x)\):
   \[
   \sin(2x) = 2 \sin(x) \cos(x)
   \]
   So the equation becomes:
   \[
   2 \sin(x) + 2 \sin(x) \cos(x) = 0
   \]
   
2. **Factor the equation**:
   \[
   2 \sin(x) (1 + \cos(x)) = 0
   \]

3. **Set each factor equal to zero**:

   - First factor: \( 2 \sin(x) = 0 \)
     \[
     \sin(x) = 0
     \]
     The solutions for \( \sin(x) = 0 \) in the interval \([0, 2\pi]\) are:
     \[
     x = 0, \pi, 2\pi
     \]
     
   - Second factor: \( 1 + \cos(x) = 0 \)
     \[
     \cos(x) = -1
     \]
     The solution for \( \cos(x) = -1 \) in the interval \([0, 2\pi]\) is:
     \[
     x = \pi
     \]
     However, this value of \(x = \pi\) is already listed from the first factor.

### Final Solution:
The solutions for the equation \( 2 \sin(x) + \sin(2x) = 0 \) in the interval \([0, 2\pi]\) are:
\[
x = 0, \pi, 2\pi
\]

Would you like more details or have any other questions?

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Here are 5 questions to consider next:
1. How can double angle identities help simplify trigonometric equations?
2. What other trigonometric identities are commonly used to solve equations?
3. What are the general solutions for \(\sin(x) = 0\) and \(\cos(x) = -1\)?
4. Can this problem be solved graphically, and what would the graph look like?
5. What would the solutions be if the interval were different, such as \([0, 4\pi]\)?

**Tip:** Factoring is a key technique in solving trigonometric equations, especially when multiple angles or identities are involved.

Find all solutions of the equation 2sin(x) + sin(2x) = 0 in the interval [0, 2π]. Write your answer in radians in terms of π.

Learn how to solve the trigonometric equation 2sin(x) + sin(2x) = 0 within the interval [0, 2π]. This step-by-step solution uses the double angle identity and factoring. Solutions are provided in radians in terms of π. Suitable for high school students.

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