If `sin2theta=(sqrt(3))/(2)` then the value of `sin3theta` is equal to (Take `0^(@)lethetale90^(@))`

Options

A. `(1)/(2)`
B. 1
C. 0
D. `(sqrt(3))/(2)`

Answer

To solve the problem, we need to find the value of \( \sin 3\theta \) given that \( \sin 2\theta = \frac{\sqrt{3}}{2} \).

### Step-by-Step Solution:

1. **Identify the angle \( 2\theta \)**:
Since \( \sin 2\theta = \frac{\sqrt{3}}{2} \), we need to determine the angles for which this is true. The sine function equals \( \frac{\sqrt{3}}{2} \) at the angles:
\[ 2\theta = 60^\circ \quad \text{or} \quad 2\theta = 120^\circ \]

2. **Solve for \( \theta \)**:
From the angles found, we can find \( \theta \):
\[ \theta = \frac{60^\circ}{2} = 30^\circ \quad \text{or} \quad \theta = \frac{120^\circ}{2} = 60^\circ \]

3. **Calculate \( \sin 3\theta \)**:
We will calculate \( \sin 3\theta \) for both values of \( \theta \).

- For \( \theta = 30^\circ \):
\[ 3\theta = 3 \times 30^\circ = 90^\circ \]
\[ \sin 3\theta = \sin 90^\circ = 1 \]

- For \( \theta = 60^\circ \):
\[ 3\theta = 3 \times 60^\circ = 180^\circ \]
\[ \sin 3\theta = \sin 180^\circ = 0 \]

4. **Conclusion**:
Therefore, the possible values of \( \sin 3\theta \) are \( 1 \) and \( 0 \) depending on the value of \( \theta \).

### Final Answer:
The value of \( \sin 3\theta \) can be either \( 1 \) or \( 0 \).