All possible values of `theta in [0, 2pi]` for which `sin 2theta +tan 2theta gt 0` lie in :

Options

A. `(0, (pi)/(2)) cup (pi, (3pi)/(2))`
B. `(0, (pi)/(2)) cup ((pi)/(2), (3pi)/(4)) cup (pi, (7pi)/(6))`
C. `(0, (pi)/(4)) cup ((pi)/(2), (3pi)/(4)) cup ((3pi)/(2), (11pi)/(6))`
D. `(0, (pi)/(4)) cup ((pi)/(2), (3pi)/(4)) cup (pi, (5pi)/(4)) cup ((3)/(2), (7pi)/(4))`

Answer

To solve the inequality \( \sin 2\theta + \tan 2\theta > 0 \) for \( \theta \) in the interval \( [0, 2\pi] \), we can follow these steps:

### Step 1: Rewrite the inequality
We know that \( \tan 2\theta = \frac{\sin 2\theta}{\cos 2\theta} \). Therefore, we can rewrite the inequality as:
\[ \sin 2\theta + \frac{\sin 2\theta}{\cos 2\theta} > 0 \]
This simplifies to:
\[ \sin 2\theta \left(1 + \frac{1}{\cos 2\theta}\right) > 0 \]
or equivalently:
\[ \sin 2\theta \cdot \frac{1 + \cos 2\theta}{\cos 2\theta} > 0 \]

### Step 2: Analyze the factors
The inequality can be analyzed by considering the signs of each factor:
1. \( \sin 2\theta > 0 \)
2. \( 1 + \cos 2\theta > 0 \) (which is always true since \( \cos 2\theta \) ranges from -1 to 1)
3. \( \cos 2\theta > 0 \)

### Step 3: Solve \( \sin 2\theta > 0 \)
The sine function is positive in the intervals:
\[ 2\theta \in (0, \pi) \cup (2\pi, 3\pi) \]
Dividing by 2 gives:
\[ \theta \in (0, \frac{\pi}{2}) \cup (\pi, \frac{3\pi}{2}) \]

### Step 4: Solve \( \cos 2\theta > 0 \)
The cosine function is positive in the intervals:
\[ 2\theta \in (-\frac{\pi}{2}, \frac{\pi}{2}) \cup (\frac{3\pi}{2}, \frac{5\pi}{2}) \]
Dividing by 2 gives:
\[ \theta \in (-\frac{\pi}{4}, \frac{\pi}{4}) \cup (\frac{3\pi}{4}, \frac{5\pi}{4}) \]

### Step 5: Find the intersection of the intervals
Now we need to find the intersection of the intervals from steps 3 and 4:
1. From \( \sin 2\theta > 0 \): \( (0, \frac{\pi}{2}) \cup (\pi, \frac{3\pi}{2}) \)
2. From \( \cos 2\theta > 0 \): \( (-\frac{\pi}{4}, \frac{\pi}{4}) \cup (\frac{3\pi}{4}, \frac{5\pi}{4}) \)

The intersections are:
- For \( (0, \frac{\pi}{2}) \) and \( (-\frac{\pi}{4}, \frac{\pi}{4}) \): The intersection is \( (0, \frac{\pi}{4}) \).
- For \( (\pi, \frac{3\pi}{2}) \) and \( (\frac{3\pi}{4}, \frac{5\pi}{4}) \): The intersection is \( (\frac{3\pi}{4}, \frac{5\pi}{4}) \).

### Step 6: Combine the results
Thus, the solution for \( \theta \) in the interval \( [0, 2\pi] \) is:
\[ \theta \in (0, \frac{\pi}{4}) \cup (\frac{3\pi}{4}, \frac{5\pi}{4}) \]

### Final Answer
The possible values of \( \theta \) for which \( \sin 2\theta + \tan 2\theta > 0 \) lie in:
\[ (0, \frac{\pi}{4}) \cup (\frac{3\pi}{4}, \frac{5\pi}{4}) \]