Let `A={theta in (-pi /2,pi):(3+2i sin theta )/(1-2 i sin theta )` is purely imaginary }
Then the sum of the elements in A is

Options

A. `(5pi)/(6)`
B. `(2pi)/(3)`
C. `(3pi)/(4)`
D. `pi`

Answer

To solve the problem, we need to find the values of \(\theta\) for which the expression \(\frac{3 + 2i \sin \theta}{1 - 2i \sin \theta}\) is purely imaginary.

### Step-by-Step Solution:

1. **Set up the expression**:
We start with the expression:
\[ z = \frac{3 + 2i \sin \theta}{1 - 2i \sin \theta} \]
We want to find when this expression is purely imaginary.

2. **Multiply numerator and denominator by the conjugate of the denominator**:
To simplify, we multiply the numerator and denominator by the conjugate of the denominator:
\[ z = \frac{(3 + 2i \sin \theta)(1 + 2i \sin \theta)}{(1 - 2i \sin \theta)(1 + 2i \sin \theta)} \]

3. **Calculate the denominator**:
The denominator simplifies as follows:
\[ (1 - 2i \sin \theta)(1 + 2i \sin \theta) = 1^2 - (2i \sin \theta)^2 = 1 + 4 \sin^2 \theta \]

4. **Calculate the numerator**:
Now, we simplify the numerator:
\[ (3 + 2i \sin \theta)(1 + 2i \sin \theta) = 3 + 6i \sin \theta + 2i \sin \theta - 4 \sin^2 \theta = 3 - 4 \sin^2 \theta + (6i \sin \theta + 2i \sin \theta) \]
This gives:
\[ = 3 - 4 \sin^2 \theta + 8i \sin \theta \]

5. **Combine the results**:
Thus, we have:
\[ z = \frac{3 - 4 \sin^2 \theta + 8i \sin \theta}{1 + 4 \sin^2 \theta} \]

6. **Separate real and imaginary parts**:
The real part \(x\) and imaginary part \(y\) of \(z\) can be expressed as:
\[ x = \frac{3 - 4 \sin^2 \theta}{1 + 4 \sin^2 \theta}, \quad y = \frac{8 \sin \theta}{1 + 4 \sin^2 \theta} \]

7. **Condition for purely imaginary**:
For \(z\) to be purely imaginary, the real part must be zero:
\[ 3 - 4 \sin^2 \theta = 0 \]

8. **Solve for \(\sin^2 \theta\)**:
Rearranging gives:
\[ 4 \sin^2 \theta = 3 \implies \sin^2 \theta = \frac{3}{4} \implies \sin \theta = \pm \frac{\sqrt{3}}{2} \]

9. **Find the angles**:
The solutions for \(\sin \theta = \frac{\sqrt{3}}{2}\) are:
\[ \theta = \frac{\pi}{3} \quad \text{and} \quad \theta = \frac{2\pi}{3} \]
The solution for \(\sin \theta = -\frac{\sqrt{3}}{2}\) is:
\[ \theta = -\frac{\pi}{3} \]

10. **Check the range**:
The values \(\frac{\pi}{3}\), \(\frac{2\pi}{3}\), and \(-\frac{\pi}{3}\) all lie within the interval \((- \frac{\pi}{2}, \pi)\).

11. **Sum the angles**:
Now, we sum the elements:
\[ \frac{\pi}{3} + \frac{2\pi}{3} - \frac{\pi}{3} = \frac{2\pi}{3} \]

### Final Answer:
The sum of the elements in \(A\) is \(\frac{2\pi}{3}\).