Find the smallest positive value of `theta` satisfying tan `theta =2 sin theta`.

Answer

To solve the equation \( \tan \theta = 2 \sin \theta \), we can follow these steps:

### Step 1: Rewrite the equation using the definition of tangent.
We know that:
\[ \tan \theta = \frac{\sin \theta}{\cos \theta} \]
So we can rewrite the equation as:
\[ \frac{\sin \theta}{\cos \theta} = 2 \sin \theta \]

### Step 2: Eliminate \(\sin \theta\) from both sides.
Assuming \(\sin \theta \neq 0\), we can multiply both sides by \(\cos \theta\) to get:
\[ \sin \theta = 2 \sin \theta \cos \theta \]

### Step 3: Rearrange the equation.
We can rearrange the equation to isolate terms:
\[ \sin \theta - 2 \sin \theta \cos \theta = 0 \]
Factoring out \(\sin \theta\):
\[ \sin \theta (1 - 2 \cos \theta) = 0 \]

### Step 4: Solve for \(\theta\).
This gives us two cases to consider:
1. \( \sin \theta = 0 \)
2. \( 1 - 2 \cos \theta = 0 \)

For the first case \( \sin \theta = 0 \):
\[ \theta = n\pi, \quad n \in \mathbb{Z} \]
This gives us values like \(0, \pi, 2\pi, \ldots\) which are not positive.

For the second case:
\[ 1 - 2 \cos \theta = 0 \implies 2 \cos \theta = 1 \implies \cos \theta = \frac{1}{2} \]
The values of \(\theta\) that satisfy \(\cos \theta = \frac{1}{2}\) are:
\[ \theta = \frac{\pi}{3} + 2n\pi \quad \text{or} \quad \theta = -\frac{\pi}{3} + 2n\pi \]
The smallest positive solution is:
\[ \theta = \frac{\pi}{3} \]

### Step 5: Convert to degrees.
To convert \(\frac{\pi}{3}\) radians to degrees:
\[ \theta = \frac{\pi}{3} \times \frac{180}{\pi} = 60^\circ \]

### Final Answer:
Thus, the smallest positive value of \(\theta\) satisfying \( \tan \theta = 2 \sin \theta \) is:
\[ \theta = 60^\circ \]