If `tan theta = 3` and `theta` lies in thrid quadrant, then the value of `sin theta` is

Options

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

Answer

To find the value of \( \sin \theta \) given that \( \tan \theta = 3 \) and \( \theta \) lies in the third quadrant, we can follow these steps:

### Step 1: Understand the relationship between trigonometric ratios
We know that:
\[ \tan \theta = \frac{\sin \theta}{\cos \theta} \]
Given \( \tan \theta = 3 \), we can express this as:
\[ \frac{\sin \theta}{\cos \theta} = 3 \]
This implies:
\[ \sin \theta = 3 \cos \theta \]

### Step 2: Use the Pythagorean identity
We also know from the Pythagorean identity that:
\[ \sin^2 \theta + \cos^2 \theta = 1 \]
Substituting \( \sin \theta = 3 \cos \theta \) into the identity gives:
\[ (3 \cos \theta)^2 + \cos^2 \theta = 1 \]
This simplifies to:
\[ 9 \cos^2 \theta + \cos^2 \theta = 1 \]
\[ 10 \cos^2 \theta = 1 \]

### Step 3: Solve for \( \cos^2 \theta \)
Now, we can solve for \( \cos^2 \theta \):
\[ \cos^2 \theta = \frac{1}{10} \]

### Step 4: Find \( \cos \theta \)
Taking the square root gives:
\[ \cos \theta = \pm \frac{1}{\sqrt{10}} \]
Since \( \theta \) is in the third quadrant, where cosine is negative:
\[ \cos \theta = -\frac{1}{\sqrt{10}} \]

### Step 5: Find \( \sin \theta \)
Now, we can find \( \sin \theta \) using \( \sin \theta = 3 \cos \theta \):
\[ \sin \theta = 3 \left(-\frac{1}{\sqrt{10}}\right) = -\frac{3}{\sqrt{10}} \]

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