If `sin theta=(1)/(3)`, hten `cos theta` will be-

Options

A. `(8)/(9)`
B. `(4)/(3)`
C. `(2sqrt2)/(3)`
D. `(3)/(4)`

Explanation

To solve the problem where we are given that \( \sin \theta = \frac{1}{3} \) and we need to find \( \cos \theta \), we can use the Pythagorean identity for sine and cosine:

### Step-by-Step Solution:

1. **Recall the Pythagorean Identity**:
The identity states that:
\[ \sin^2 \theta + \cos^2 \theta = 1 \]

2. **Substitute the value of \( \sin \theta \)**:
We know that \( \sin \theta = \frac{1}{3} \). Therefore, we can calculate \( \sin^2 \theta \):
\[ \sin^2 \theta = \left(\frac{1}{3}\right)^2 = \frac{1}{9} \]

3. **Plug \( \sin^2 \theta \) into the identity**:
Now substitute \( \sin^2 \theta \) into the Pythagorean identity:
\[ \frac{1}{9} + \cos^2 \theta = 1 \]

4. **Isolate \( \cos^2 \theta \)**:
Rearranging the equation gives:
\[ \cos^2 \theta = 1 - \frac{1}{9} \]

5. **Simplify the right-hand side**:
To simplify \( 1 - \frac{1}{9} \), we convert 1 into a fraction:
\[ 1 = \frac{9}{9} \]
Thus,
\[ \cos^2 \theta = \frac{9}{9} - \frac{1}{9} = \frac{8}{9} \]

6. **Take the square root**:
Now, we take the square root of both sides to find \( \cos \theta \):
\[ \cos \theta = \pm \sqrt{\frac{8}{9}} = \pm \frac{\sqrt{8}}{3} = \pm \frac{2\sqrt{2}}{3} \]

7. **Select the appropriate value**:
Since the options provided are positive, we take the positive value:
\[ \cos \theta = \frac{2\sqrt{2}}{3} \]

### Final Answer:
Thus, the value of \( \cos \theta \) is:
\[ \cos \theta = \frac{2\sqrt{2}}{3} \]