If `cosx=-(3)/(5)andpiltxlt(3pi)/(2)`,then the value of sin2x will be

Options

A. `(12)/(25)`
B. `(1)/(15)`
C. `(24)/(25)`
D. `(5)/(26)`

Answer

To solve the problem, we need to find the value of \( \sin 2x \) given that \( \cos x = -\frac{3}{5} \) and \( \pi < x < \frac{3\pi}{2} \).

### Step-by-Step Solution:

1. **Identify the Quadrant**:
Since \( x \) is between \( \pi \) and \( \frac{3\pi}{2} \), it is in the third quadrant. In this quadrant, both sine and cosine are negative.

2. **Use the Pythagorean Identity**:
We know that:
\[ \sin^2 x + \cos^2 x = 1 \]
Given \( \cos x = -\frac{3}{5} \), we can substitute this value into the identity:
\[ \sin^2 x + \left(-\frac{3}{5}\right)^2 = 1 \]
This simplifies to:
\[ \sin^2 x + \frac{9}{25} = 1 \]

3. **Solve for \( \sin^2 x \)**:
Rearranging the equation gives:
\[ \sin^2 x = 1 - \frac{9}{25} \]
Converting 1 into a fraction with a denominator of 25:
\[ \sin^2 x = \frac{25}{25} - \frac{9}{25} = \frac{16}{25} \]

4. **Find \( \sin x \)**:
Taking the square root of both sides, we have:
\[ \sin x = \pm \sqrt{\frac{16}{25}} = \pm \frac{4}{5} \]
Since \( x \) is in the third quadrant, \( \sin x \) must be negative:
\[ \sin x = -\frac{4}{5} \]

5. **Calculate \( \sin 2x \)**:
We use the double angle formula for sine:
\[ \sin 2x = 2 \sin x \cos x \]
Substituting the values we have:
\[ \sin 2x = 2 \left(-\frac{4}{5}\right) \left(-\frac{3}{5}\right) \]
This simplifies to:
\[ \sin 2x = 2 \cdot \frac{12}{25} = \frac{24}{25} \]

### Final Answer:
Thus, the value of \( \sin 2x \) is:
\[ \sin 2x = \frac{24}{25} \]