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))`

Explanation

To solve the problem where \( \tan \theta = 3 \) and \( \theta \) lies in the third quadrant, we will follow these steps:

### Step-by-Step Solution:

1. **Understanding the Quadrant**:
Since \( \theta \) is in the third quadrant, both \( \tan \theta \) and \( \cos \theta \) are positive, while \( \sin \theta \) is negative.

2. **Using the Tangent Identity**:
Given \( \tan \theta = 3 \), we can express this in terms of sine and cosine:
\[ \tan \theta = \frac{\sin \theta}{\cos \theta} = 3 \]
This implies:
\[ \sin \theta = 3 \cos \theta \]

3. **Using the Pythagorean Identity**:
We know from the Pythagorean identity that:
\[ \sin^2 \theta + \cos^2 \theta = 1 \]
Substituting \( \sin \theta = 3 \cos \theta \) into this identity gives:
\[ (3 \cos \theta)^2 + \cos^2 \theta = 1 \]
Simplifying this, we have:
\[ 9 \cos^2 \theta + \cos^2 \theta = 1 \]
\[ 10 \cos^2 \theta = 1 \]
\[ \cos^2 \theta = \frac{1}{10} \]

4. **Finding Cosine**:
Taking the square root gives:
\[ \cos \theta = \pm \sqrt{\frac{1}{10}} = \pm \frac{1}{\sqrt{10}} \]
Since \( \theta \) is in the third quadrant, \( \cos \theta \) is negative:
\[ \cos \theta = -\frac{1}{\sqrt{10}} \]

5. **Finding Sine**:
Now, we can find \( \sin \theta \) using the relationship \( \sin \theta = 3 \cos \theta \):
\[ \sin \theta = 3 \left(-\frac{1}{\sqrt{10}}\right) = -\frac{3}{\sqrt{10}} \]

### Final Answer:
Thus, the value of \( \sin \theta \) is:
\[ \sin \theta = -\frac{3}{\sqrt{10}} \]