If `cot^(-1) ( - (1)/(sqrt(3)))` =x then the value of sin x is

Options

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

Answer

To solve the problem, we need to find the value of \( \sin x \) given that \( x = \cot^{-1} \left( -\frac{1}{\sqrt{3}} \right) \).

### Step-by-Step Solution:

1. **Understanding the Inverse Cotangent**:
We start with the expression \( x = \cot^{-1} \left( -\frac{1}{\sqrt{3}} \right) \). The cotangent function is negative in the second quadrant.

2. **Using the Cotangent Identity**:
We can use the identity for the cotangent of a negative angle:
\[ \cot^{-1}(-\theta) = \pi - \cot^{-1}(\theta) \]
Thus, we can rewrite our expression:
\[ x = \pi - \cot^{-1} \left( \frac{1}{\sqrt{3}} \right) \]

3. **Finding \( \cot^{-1} \left( \frac{1}{\sqrt{3}} \right) \)**:
We know that \( \cot \left( \frac{\pi}{3} \right) = \frac{1}{\sqrt{3}} \). Therefore:
\[ \cot^{-1} \left( \frac{1}{\sqrt{3}} \right) = \frac{\pi}{3} \]

4. **Substituting Back to Find \( x \)**:
Now substituting this back into our equation for \( x \):
\[ x = \pi - \frac{\pi}{3} \]
To simplify this, we can express \( \pi \) as \( \frac{3\pi}{3} \):
\[ x = \frac{3\pi}{3} - \frac{\pi}{3} = \frac{2\pi}{3} \]

5. **Calculating \( \sin x \)**:
Now we need to find \( \sin x \):
\[ \sin x = \sin \left( \frac{2\pi}{3} \right) \]
We can use the sine identity:
\[ \sin \left( \frac{2\pi}{3} \right) = \sin \left( \pi - \frac{\pi}{3} \right) = \sin \left( \frac{\pi}{3} \right) \]
Since \( \sin(\pi - \theta) = \sin(\theta) \).

6. **Final Value of \( \sin \left( \frac{\pi}{3} \right) \)**:
We know that:
\[ \sin \left( \frac{\pi}{3} \right) = \frac{\sqrt{3}}{2} \]

### Conclusion:
Thus, the value of \( \sin x \) is:
\[ \sin x = \frac{\sqrt{3}}{2} \]