If `sintheta=(1)/(3)`,then find the value of `sintheta*cos^(2)theta+"cosec"theta`.

Answer

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

### Step-by-Step Solution:

1. **Identify the given value**:
\[ \sin \theta = \frac{1}{3} \]

2. **Find \( \cos^2 \theta \)**:
We can use the Pythagorean identity:
\[ \cos^2 \theta = 1 - \sin^2 \theta \]
First, calculate \( \sin^2 \theta \):
\[ \sin^2 \theta = \left(\frac{1}{3}\right)^2 = \frac{1}{9} \]
Now, substitute this into the identity:
\[ \cos^2 \theta = 1 - \frac{1}{9} = \frac{9}{9} - \frac{1}{9} = \frac{8}{9} \]

3. **Calculate \( \sin \theta \cdot \cos^2 \theta \)**:
Now we can find \( \sin \theta \cdot \cos^2 \theta \):
\[ \sin \theta \cdot \cos^2 \theta = \frac{1}{3} \cdot \frac{8}{9} = \frac{8}{27} \]

4. **Find \( \csc \theta \)**:
The cosecant function is the reciprocal of the sine function:
\[ \csc \theta = \frac{1}{\sin \theta} = \frac{1}{\frac{1}{3}} = 3 \]

5. **Combine the two results**:
Now, we can combine the results from steps 3 and 4:
\[ \sin \theta \cdot \cos^2 \theta + \csc \theta = \frac{8}{27} + 3 \]
To add these, convert 3 into a fraction with a denominator of 27:
\[ 3 = \frac{3 \cdot 27}{27} = \frac{81}{27} \]
Now, add the fractions:
\[ \frac{8}{27} + \frac{81}{27} = \frac{8 + 81}{27} = \frac{89}{27} \]

6. **Final Result**:
Therefore, the final answer is:
\[ \frac{89}{27} \]