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

Options

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

Explanation

To solve the problem, we need to determine the values of \(\theta\) such that the expression \(\frac{3 + 2i \sin \theta}{1 - 2i \sin \theta}\) is purely imaginary. This means that the real part of the expression must equal zero.

### Step-by-Step Solution:

1. **Set up the equation**: We have the expression \(\frac{3 + 2i \sin \theta}{1 - 2i \sin \theta}\). We want to find when this is purely imaginary, meaning the real part must be zero.

2. **Rationalize the denominator**: Multiply the numerator and denominator by the conjugate of the denominator:
\[ \frac{(3 + 2i \sin \theta)(1 + 2i \sin \theta)}{(1 - 2i \sin \theta)(1 + 2i \sin \theta)}. \]

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

4. **Calculate the numerator**:
\[ (3 + 2i \sin \theta)(1 + 2i \sin \theta) = 3 + 6i \sin \theta + 2i \sin \theta + 4i^2 \sin^2 \theta = 3 + 8i \sin \theta - 4 \sin^2 \theta. \]
Thus, the numerator simplifies to:
\[ (3 - 4 \sin^2 \theta) + 8i \sin \theta. \]

5. **Combine the results**: The expression now becomes:
\[ \frac{(3 - 4 \sin^2 \theta) + 8i \sin \theta}{1 + 4 \sin^2 \theta}. \]

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

7. **Set the real part to zero**: For the expression to be purely imaginary, we set \(x = 0\):
\[ 3 - 4 \sin^2 \theta = 0. \]

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

9. **Determine the angles**: The values of \(\theta\) for \(\sin \theta = \frac{\sqrt{3}}{2}\) are:
\[ \theta = \frac{\pi}{3}, \quad \text{and for } \sin \theta = -\frac{\sqrt{3}}{2}, \quad \theta = -\frac{\pi}{3}. \]

10. **Check the range**: The angles must be in the interval \((- \frac{\pi}{2}, \pi)\). Both \(\frac{\pi}{3}\) and \(-\frac{\pi}{3}\) are valid.

11. **Sum the elements**: The sum of the elements in set \(A\) is:
\[ \frac{\pi}{3} + \left(-\frac{\pi}{3}\right) + \frac{2\pi}{3} = \frac{2\pi}{3}. \]

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