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The equation provided is:

\[
2 \sin x - \sin 2x = 0
\]

We need to solve this equation in the interval \([0, 2\pi]\).

### Step 1: Simplify the equation using trigonometric identities

We know that \(\sin(2x) = 2\sin(x)\cos(x)\). Using this identity, the equation becomes:

\[
2\sin x - 2\sin x \cos x = 0
\]

### Step 2: Factor the equation

Factor out \(2\sin x\) from both terms:

\[
2\sin x (1 - \cos x) = 0
\]

### Step 3: Solve each factor

There are two factors to solve:

1. \(2\sin x = 0\)
2. \(1 - \cos x = 0\)

#### Case 1: \(2\sin x = 0\)

\[
\sin x = 0
\]

In the interval \([0, 2\pi]\), \(\sin x = 0\) at \(x = 0, \pi, 2\pi\).

#### Case 2: \(1 - \cos x = 0\)

\[
\cos x = 1
\]

In the interval \([0, 2\pi]\), \(\cos x = 1\) at \(x = 0\).

### Step 4: Combine the solutions

The solutions from both cases are \(x = 0, \pi, 2\pi\). However, since the interval is \([0, 2\pi]\) (closed interval including 0 and \(2\pi\)), all values are valid.

### Final Answer:

\[
x = 0, \pi, 2\pi
\]

Would you like more details, or do you have any questions?

---

Here are 5 related questions to further your understanding:

1. How does the double angle identity for sine help simplify trigonometric equations?
2. What is the general solution for \(\sin x = 0\) and \(\cos x = 1\) in other intervals?
3. Can you solve a similar equation involving \(\sin 3x\) instead of \(\sin 2x\)?
4. How would the solution change if the interval was \([0, \pi]\) instead of \([0, 2\pi]\)?
5. What other types of trigonometric identities can simplify equations like this?

**Tip**: Whenever you see a product of trigonometric functions set to zero, remember to solve each factor separately for the possible solutions.

Find all solutions of the equation 2 sin x - sin 2x = 0 in the interval [0, 2π].

Learn how to solve the trigonometric equation 2 sin x - sin 2x = 0 using trigonometric identities and the zero product property. This solution covers step-by-step simplifications and identifies all solutions in the interval [0, 2π]. Suitable for students in Grades 10-12.

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