If `tantheta=3` and `theta` lies in third quadrant then `sintheta=`

Options

A. `1/sqrt(10)`
B. `-1/sqrt(10)`
C. `-3/sqrt(10)`
D. `3/sqrt(10)`

Answer

To solve the problem, we need to find the value of \(\sin \theta\) given that \(\tan \theta = 3\) and \(\theta\) lies in the third quadrant.

### Step-by-step Solution:

1. **Understand the relationship of tangent**:
\[ \tan \theta = \frac{\text{opposite}}{\text{adjacent}} = \frac{3}{1} \]
Here, we can denote the opposite side as 3 and the adjacent side as 1.

2. **Use the Pythagorean theorem to find the hypotenuse**:
\[ \text{hypotenuse} = \sqrt{(\text{opposite})^2 + (\text{adjacent})^2} = \sqrt{3^2 + 1^2} = \sqrt{9 + 1} = \sqrt{10} \]

3. **Find \(\sin \theta\)**:
\[ \sin \theta = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{3}{\sqrt{10}} \]

4. **Determine the sign of \(\sin \theta\)**:
Since \(\theta\) is in the third quadrant, where sine is negative, we have:
\[ \sin \theta = -\frac{3}{\sqrt{10}} \]

5. **Final answer**:
\[ \sin \theta = -\frac{3}{\sqrt{10}} \]