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We are tasked with solving the equation:

\[
\sin(2\theta) = \tan(\theta)
\]

### Step 1: Express \(\tan(\theta)\) in terms of sine and cosine

We know that:

\[
\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}
\]

Substituting this into the original equation:

\[
\sin(2\theta) = \frac{\sin(\theta)}{\cos(\theta)}
\]

### Step 2: Use the double-angle identity for \(\sin(2\theta)\)

The double-angle identity for sine is:

\[
\sin(2\theta) = 2\sin(\theta)\cos(\theta)
\]

Substitute this into the equation:

\[
2\sin(\theta)\cos(\theta) = \frac{\sin(\theta)}{\cos(\theta)}
\]

### Step 3: Eliminate \(\sin(\theta)\)

If \(\sin(\theta) \neq 0\), we can divide both sides of the equation by \(\sin(\theta)\):

\[
2\cos(\theta) = \frac{1}{\cos(\theta)}
\]

### Step 4: Solve for \(\cos(\theta)\)

Multiply both sides by \(\cos(\theta)\) to clear the fraction:

\[
2\cos^2(\theta) = 1
\]

Divide both sides by 2:

\[
\cos^2(\theta) = \frac{1}{2}
\]

Taking the square root of both sides:

\[
\cos(\theta) = \pm \frac{1}{\sqrt{2}} = \pm \frac{\sqrt{2}}{2}
\]

### Step 5: Determine the values of \(\theta\)

The general solutions for \(\cos(\theta) = \pm \frac{\sqrt{2}}{2}\) are:

\[
\theta = \frac{\pi}{4} + 2n\pi \quad \text{or} \quad \theta = \frac{3\pi}{4} + 2n\pi
\]
where \(n\) is any integer.

### Step 6: Consider the case when \(\sin(\theta) = 0\)

If \(\sin(\theta) = 0\), then:

\[
\theta = n\pi \quad \text{for} \quad n \in \mathbb{Z}
\]

### Final Solution

The complete set of solutions is:

\[
\theta = n\pi \quad \text{or} \quad \theta = \frac{\pi}{4} + 2n\pi \quad \text{or} \quad \theta = \frac{3\pi}{4} + 2n\pi
\]
where \(n\) is any integer.

Would you like further details on any of these steps?

---

Here are 5 related questions to expand your understanding:
1. How do you derive the double-angle identity for sine?
2. What are the general solutions for \(\sin(\theta) = 0\)?
3. How would the solution change if the equation involved \(\cos(2\theta)\) instead of \(\sin(2\theta)\)?
4. Can you graphically interpret the solutions on the unit circle?
5. How do solutions change if we limit \(\theta\) to a specific interval, like \([0, 2\pi]\)?

**Tip:** When solving trigonometric equations, always check for extraneous solutions by considering cases where the trigonometric functions could be undefined.

Compute all the angles θ such that sin(2θ) = tan(θ)

Learn how to solve the trigonometric equation sin(2θ) = tan(θ) with detailed step-by-step instructions. This solution involves using double angle identities and understanding the general solutions for trigonometric functions. Suitable for advanced high school students in Grades 11-12.

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