If `sin theta=(1)/(2), " write the value of " (3cot^(2)theta+3).`

Explanation

To solve the problem, we need to find the value of \(3 \cot^2 \theta + 3\) given that \(\sin \theta = \frac{1}{2}\).

### Step-by-Step Solution:

1. **Find \(\cot^2 \theta\)**:
We know that:
\[ \cot \theta = \frac{\cos \theta}{\sin \theta} \]
Therefore,
\[ \cot^2 \theta = \frac{\cos^2 \theta}{\sin^2 \theta} \]

2. **Calculate \(\sin^2 \theta\)**:
Given \(\sin \theta = \frac{1}{2}\),
\[ \sin^2 \theta = \left(\frac{1}{2}\right)^2 = \frac{1}{4} \]

3. **Use the Pythagorean identity to find \(\cos^2 \theta\)**:
The Pythagorean identity states:
\[ \sin^2 \theta + \cos^2 \theta = 1 \]
Substituting \(\sin^2 \theta\):
\[ \frac{1}{4} + \cos^2 \theta = 1 \]
Solving for \(\cos^2 \theta\):
\[ \cos^2 \theta = 1 - \frac{1}{4} = \frac{3}{4} \]

4. **Substitute \(\cos^2 \theta\) and \(\sin^2 \theta\) into \(\cot^2 \theta\)**:
Now we can find \(\cot^2 \theta\):
\[ \cot^2 \theta = \frac{\cos^2 \theta}{\sin^2 \theta} = \frac{\frac{3}{4}}{\frac{1}{4}} = 3 \]

5. **Calculate \(3 \cot^2 \theta + 3\)**:
Now we can substitute \(\cot^2 \theta\) into the expression:
\[ 3 \cot^2 \theta + 3 = 3 \cdot 3 + 3 = 9 + 3 = 12 \]

### Final Answer:
Thus, the value of \(3 \cot^2 \theta + 3\) is \(12\).

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